TEKREVOLUTION is a high-tech service and consulting company from Italy that has extensive experience in R&D activities. One of their main activities is to explore intelligent additive manufacturing to provide higher stress-to-weight ratio solutions for the market.

## Challenge
This use case is part of a bigger project done together with TekRevoultion and ESA to demonstrate the capabilities of Uptim.ai software to help during the optimization in topology optimization processes. The primary goal was to increase the robustness of the optimization, particularly for multiobjective purposes, in order to get a geometry performing well under different loads. The secondary objective was to reduce the manual hours of engineering experts when optimizing, reducing the complexity and the number of operations with a special focus on the postprocessing part.
In this case, we will study an antenna bracket that substituted an original optimization use case (Vega C IS 2-3 FWD support). The allowable volume and the requirements for optimization were given by ESA.

![antenna_bracket_1.jpg](https://uptimai-website.s3.eu-central-1.amazonaws.com/antenna_bracket_1_86c52fd84e.jpg)

*Figure 1 – Computational mesh of the allowable volume of the antenna bracket (coarse mesh)*

In particular, the mechanical requirements are the following:
- 1st main mode > 140 Hz
- Quasi-static loads: 40g in all directions
- 4 Ifs with the unit, modeled as a 0.4 kg concentrated mass at the center point of the holes


## Solution
The Statistical Optimization feature works by generating iteratively surrogate models while reducing the domain of study to where the zone of interest (zone of optimum) is located. That way, the model keeps getting refined only where the maximum probability of having a good solution is located.
This approach has two main benefits in comparison to standard optimization techniques. The first benefit is that the number of samples needed until convergence is much lower, resulting in a decreased computational cost. The second benefit is that as the result is not a single point, but a range of possible values, the solution provided is much more robust to final adjustments and changes.

In this case, a first FE model is realized using the Inspire interface (Figure 3.1). The input parameters used for this problem are:
- MEMBSIZ: The smallest size of an element of the structure
- STRESS: Maximum stress supported by each element of the structure
- theta & fi: Normalised normal vector of deposition plane in spherical coordinates

And the inquired outputs are:

- WCOMP: Weighted compliance index (minimized during topology optimization)
- MASS: Target mass
- FREQ: 1st mode frequency

Uptim.ai Statistical optimization was used to minimize the mass while being compliant with the constraint of the first natural frequency. Three iterations (i.e. models) were realized by Uptim SO. The simulations were done by coupling our software with OptiStruct, so we were automatically changing the input files, running the Optistruct, and reading its results to adaptively choose where the next sample would be in order to generate the model as efficiently as possible.
In the following pictures, it can be seen the domain reduction for some of the inputs (the angles of the vector of deposition) into the zone where the optimum is located, iteration by iteration, as well as the behaviour of the mass (final main output).

![antenna_bracket_2.png](https://uptimai-website.s3.eu-central-1.amazonaws.com/antenna_bracket_2_7f0f7bfaab.png)

*Figure 2 – FI input range convergence*

![antenna_bracket_3.png](https://uptimai-website.s3.eu-central-1.amazonaws.com/antenna_bracket_3_3c4d8ba7f8.png)

*Figure 3 – THETA input range convergence*

![antenna_bracket_4.png](https://uptimai-website.s3.eu-central-1.amazonaws.com/antenna_bracket_4_88e767d9dd.png)

*Figure 4 – MASS results convergence*

As we can see in the previous pictures, we end up with a narrow range of possible solutions of the angles of deposition that end up with a considerable decrease in the final mass of the component. No matter which is the exact value inside that range, the mass that we will have will be in the green distribution of the last figure, and being well optimized.
In general, we can see as well how the mass changes depending on the two angles of deposition, with the clean minimum zone identified.

![antenna_bracket_5.png](https://uptimai-website.s3.eu-central-1.amazonaws.com/antenna_bracket_5_017da3dc47.png)

*Figure 5 – theta and fi detailed model*

The optimum zone was consistent with the frequency requirement, as all the combinations of inputs led to a first natural mode higher than 140 Hz (viable solutions). So there was no real constraint on whether the angles of deposition were actually valid or not from the frequency side (Figure below).

![antenna_bracket_6.png](https://uptimai-website.s3.eu-central-1.amazonaws.com/antenna_bracket_6_ed2b611278.png)

*Figure 6 – Frequency convergence (theta and fi)*

Nevertheless, Optistruct was not able to converge from the stress side in the whole domain, finding a zone (where the Stress was lower to 21) where convergence was not guaranteed. There were still some points converging, but if chosen the optimum point with the stress inside the 20 - 21 range, a deeper analysis should be made to ensure convergence.

![antenna_bracket_7.png](https://uptimai-website.s3.eu-central-1.amazonaws.com/antenna_bracket_7_7cd3d9d06c.png)

*Figure 7 – Global convergence (stress)*

The input distributions for the final iteration (Figure 3.3 to 3.6) led to a mass minimization range. The final optimized configuration was found to be 2.2 kg. The result in terms of the final net shape with the coarse initial mesh is shown in the following figure.

![antenna_bracket_8.jpg](https://uptimai-website.s3.eu-central-1.amazonaws.com/antenna_bracket_8_0eabff43c4.jpg)

*Figure 8 – Final net shape of antenna bracket (coarse mesh)*

The mesh was then refined and a more detailed net shape was obtained and smoothed with Inspire, with the parameters suggested by our software. In the final printable geometry, a last simulation is made in order to verify that all the requirements are met.

![antenna_bracket_9.jpg](https://uptimai-website.s3.eu-central-1.amazonaws.com/antenna_bracket_9_a4f5d2ea7c.jpg)

*Figure 9 – Final net shape of antenna bracket (refined mesh)*

## Benefits
- **Reduction of the mass up to 9.6%** (in comparison to using direct topology optimization from Optistruct)
- **Increase in performance** (frequency, stress and displacement)
- Ensuring **robustness** in the design
- **Reducing the cost** of postprocessing


small_antenna_bracket_title.png

thumbnail_antenna_bracket_title.png

antenna_bracket_title.png

Antenna Bracket Design

This project was developed with a US company in the defense sector and was about creating a material based on nanotubes for armoring. **Because of confidentiality in this document, the inputs and outputs won't be named, as well as the data and models have been normalized, simplified, and slightly modified.**

## Challenge
Material creation in laboratories is a multifaceted process, where a variety of inputs must be carefully controlled and optimized to achieve desired outcomes. One of the primary factors that influence the final material properties is the selection of raw materials. The purity, composition, and even the source of the materials can play a pivotal role in determining the final product’s characteristics. Enriching base materials by adding specific additives or dopants—such as rare earth metals in semiconductors or carbon nanotubes in polymers—can enhance particular properties like strength, electrical conductivity, or thermal stability. The process of introducing these enriching agents must be precisely controlled, as even slight variations in concentration can result in significant differences in performance, particularly in advanced materials like high-performance alloys, biomaterials, or nanomaterials.

Equally important are the physical conditions under which materials are synthesized. Temperature, pressure, and reaction time can all influence the formation and microstructure of the material, potentially enriching its properties. For example, by varying the temperature during a synthesis reaction, researchers can influence the crystallization rate, which in turn can enhance the material’s mechanical or electrical properties. High-pressure environments are often used to simulate extreme conditions that materials may encounter in real-world applications, leading to the creation of enriched, high-density materials with superior strength or resilience. Furthermore, innovative techniques like high-energy ball milling or laser-assisted deposition can alter the morphology of materials on the nanoscale, enriching their performance in ways that conventional methods cannot achieve.

The fabrication method also plays a crucial role in enriching material properties. Techniques like additive manufacturing (3D printing), chemical vapor deposition (CVD), or sol-gel processing enable precise control over material structure at the atomic or molecular level. These methods can incorporate enriched components, such as nanoparticles or fibers, into matrices to improve properties like strength, conductivity, or corrosion resistance. By tailoring the microstructure during fabrication, researchers can create advanced composites or functional materials that offer superior performance for specific applications, from aerospace to biomedical devices.

![Nanotubes_01.png](https://uptimai-website.s3.eu-central-1.amazonaws.com/Nanotubes_01_2f9e9ca2c7.png)

For this study, multiple variables (>60) have been taken into account to study 9 different relevant material characteristics in order to understand how they are affecting the material and how its performance could be optimized from a multi-objective perspective.

Target: 
- Find a model that can track the properties of the material
- Find the right combination of additives to fulfill the required properties
- Find the multidisciplinary optimal point 
- Suggest new measurements to improve the quality of the final model
- Reduce the cost of production if possible


## Solution
In this case, a very small set of experiments in the laboratory were already performed and the objective was to find which new combinations of variables would be of high interest for a new testing campaign in order to maximize the probability of a better material finding. That way it was a problem for the AI modeller and only data was provided. 

Uptimai tool is used to read the data (in the form of .csv table) and create a digital twin of the material itself, giving insights into the problem. The first thing to do after creating the model is validate it, to understand the expected error order of magnitude that we are taking into account.

![Nanotubes_02.png](https://uptimai-website.s3.eu-central-1.amazonaws.com/Nanotubes_02_6d80174238.png)

In this case, we can see that even having a very small dataset Uptimai is able to find a proper Digital twin model that is actually explaining the behaviour of the problem. The error is randomly distributed against all relevant variables and against the output, showing that the model is properly fitted in the whole domain, i.e. we have an accurate model using a very limited number of samples.

![Nanotubes_03.png](https://uptimai-website.s3.eu-central-1.amazonaws.com/Nanotubes_03_829197d5a3.png)

The second thing that was done, once the models were ensured to be accurate, was to do a sensitivity analysis (compare the importance). This shall allow us to target additive parts, which can be replaced with cheaper solutions as they will not impact the studied property of the problem.

Three variables were the most relevant, two being the quantity of presence of an additive and one a manufacturing process. It turns out that high-level interaction played an important role in the process and thus, it was suggested to study this interaction closer, if the production cost would play an important role.

![Nanotubes_04.png](https://uptimai-website.s3.eu-central-1.amazonaws.com/Nanotubes_04_1050991d51.png)

To create a statistical analysis, one needs a large portion of samples. Our algorithm created distributions using only a limited number of samples allowing for inter-outcome statistical analysis. 

![Nanotubes_05.png](https://uptimai-website.s3.eu-central-1.amazonaws.com/Nanotubes_05_a215089c3a.png)

Checking the posterior distribution based on the few data (distributions generated using our algorithm) showed that there is the possibility to obtain good results. In other words, the thick distribution on the right side (Figure above) suggests that the found optimum shall be stable and not be sensitive to production tolerances and additive impurities.

Using our statistical tooling for maximal and minimal regions (Figure below), we quickly established the bounds for each variable. It can be easily seen that being in the lower domain we get a high probability of high outcome. Easily, we can establish bounds on the additives, i.e. how the range for additive material shall be from 0 to 0.2. 

![Nanotubes_06.png](https://uptimai-website.s3.eu-central-1.amazonaws.com/Nanotubes_06_cde1f8a9f9.png)

![Nanotubes_07.png](https://uptimai-website.s3.eu-central-1.amazonaws.com/Nanotubes_07_23068d1817.png)

Interesting statistical analysis between various outputs shows a strong correlation between outputs 7 and 6. With an interesting point of stability around higher values of output 7 and output 6. Studying this desired spot allows us to tune the material according to the needs of the customer and simultaneously make the material cheaper to produce. In order to make the final solution cheaper, we do not aim for the maximum values of outcome 6 but slightly less. This allowed us to find a stable solution that is insensitive to outside influences, e.g. production tolerances. 

![Nanotubes_08.png](https://uptimai-website.s3.eu-central-1.amazonaws.com/Nanotubes_08_576a5e050a.png)

From Figure above, we can see a strong correlation between outcomes 7 and 6. It allowed us to establish design flexibility for each outcome. In other words, outcomes 7 and 6 have a sweet spot around 0.23 for variable 2, but to prevent a low outcome for output 7, one has to make sure that values of variable 2 are not below 0.02, i.e. close to zero and also above 0.5 as it would steadily decrease performance for both outcomes. 

Lastly in this case it was important to find how the relevant variables were affecting the different outputs. For the first step, there are important directions and potential gradients of the domain. 

For output 3 (Figure below), it is found that the interaction between variables 1 and 2 forms a plateau where the output remains constant, except for a specific zone where both variables are set to zero, where there is a huge decrease in the performance of the material. It is necessary to avoid that zone at all costs, as it creates an unstable performance and potential customer complaints on quality.

![Nanotubes_09.png](https://uptimai-website.s3.eu-central-1.amazonaws.com/Nanotubes_09_7257df8ff8.png)

Also for output 3, but for variables 2 and 3, it can be seen that for a better performance of the material, variables 2 and 3 (which are related to the addition of two different additives), have a direct correlation, needing that both additives are added at the same proportion to have a positive impact on the result. This simplified the production process as measuring the same portions is easier to track. 

![Nanotubes_10.png](https://uptimai-website.s3.eu-central-1.amazonaws.com/Nanotubes_10_d1fc90b0ee.png)

Finally, for Output 4, there was an important interaction between 3 variables that indicated the position of an optimum in a very specific part of the domain, where variable 4 was in the range of 3 to 4.

![Nanotubes_11.png](https://uptimai-website.s3.eu-central-1.amazonaws.com/Nanotubes_11_507ac31dfb.png)

With the crossed information of all outputs, we have a desired zone where to find the optimum. New tests following Uptimai indications ended up with the desired material found with a very low number of lab tests.

## Benefits
- **Understanding of the physics** behind the problem
- **Reduction of cost and time** during the R&D processes Fewer laboratory tests)
- **Increase the performance** of the material, as a better optimum is found
- **Multidisciplinary optimum** analysis
- **Stable region obtained** with stable outcomes, i.e. insensitive to production tolerances
- Ways to **decrease production costs**


thumbnail_Nanotubes_01.png

Nanotubes_01.png

Nanotubes material

CONTINENTAL AUTOMOTIVE CZECH REPUBLIC is a major Tier-1 supplier for the automotive industry. To maintain the top quality of its products, the company incorporates progressive methods of testing and analysis of measured characteristics.

## Challenge
The vast majority of modern passenger cars are equipped with an electronic parking brake allowing the use of hill assist control and similar technologies. The braking unit consists of the electric motor and the worm drive gearbox. The manufacturing of gears is crucial for the efficiency of the drive, which also includes the noise generated by the operation of the parking brake. The goal is of course to minimize the unwanted noise from applying/releasing the parking brake.

From the engineering experience, it is obvious that the noise level depends on the final manufacturing of gears in the braking unit. However, first, the main source has to be identified in the set of gear geometry characteristics. Then, a proper combination of manufacturing tolerances needs to be found to reduce the noise at optimal costs. The lengthy process of precise measurement of the gearing is limiting the number of collected data available for analysis.

<img 
 style="display: block; 
 margin-left: auto;
 margin-right: auto;
 width: 50%;
 text-align: center;"
 src="https://uptimai-website.s3.eu-central-1.amazonaws.com/pebrake_control_crop_8857c7b10f.jpg" 
 alt="pebrake-control-crop">
</img>
Fig. 1. Electronic parking brake control

## Solution
At the start, a review of measured data can give first valuable information about the solved task. The **Uptimai preprocessing tool** allows the visualization of data points in a variety of adjustable plots. Here, it discovered phenomena such as clustering of measurements or certain types of input variable correlations. Moreover, it helped to identify outliers, from the point of view of input parameter ranges as well as in terms of output values. Also, the data review served to set the method of the **Uptimai algorithm** correctly, since it had to build a reliable mathematical model from only 50 measured samples when considering 16 input variables**!** The algorithm was able to achieve the overall precision of the model by approximately 90%.

<img 
 style="display: block; 
 margin-left: auto;
 margin-right: auto;
 width: 50%;
 text-align: center;"
 src="https://uptimai-website.s3.eu-central-1.amazonaws.com/scatter_variable_4_variable_5_clean_953cd098ff.png" 
 alt="scatter_variable_4_variable_5_clean">
</img>

<img 
 style="display: block; 
 margin-left: auto;
 margin-right: auto;
 width: 50%;
 text-align: center;"
 src="https://uptimai-website.s3.eu-central-1.amazonaws.com/scatter_output_1_output_3_clean_baf8748067.png" 
 alt="scatter_output_1_output_3_clean">
</img>
 Fig. 2. Review of measured source data

When the surrogate model of the problem was ready, the approach to its analysis was done in the **Uptimai postprocessing tool** the standard way. Step-by-step it was investigated in all available features of the software just like models that originate not from measured data, but are based on the uncertainty quantification approach coupled with simulation software. Some parameters of the main-gear geometry were identified as the most crucial for the noise. The important finding was that interactions between these parameters cannot be omitted and have to be considered when solving the noise issue.

The model was able to describe the main trends in dependencies between examined outputs and geometric characteristics of the gear, a.k.a. input variables. Based on these, **Regions of minimum/maximum** were computed for each variable. On each plot, there can be seen immediately where a design point is located in the range of an input variable, where the probability of the increase in unwanted noise is higher (green area) and where the chance for lower noise of the gear prevails (red area). These plots were generated for each output in the analysis and with the simple overlay of such plots engineers were able to find viable trade-offs in tolerance fields of all geometric characteristics.

<img 
 style="display: block; 
 margin-left: auto;
 margin-right: auto;
 width: 50%;
 text-align: center;"
 src="https://uptimai-website.s3.eu-central-1.amazonaws.com/rmm_input_df_clean_34838722cb.png" 
 alt="rmm_input_df_clean">
</img>
Fig. 3. Review of measured source data

Speaking about trade-offs, there is also a possibility to directly compare model results of different outputs against each other. Here it was confirmed that noise levels with- and without loading of the gear mechanism are tightly correlated. Moreover, clear trends in behaviour based on the most important inputs were also observed from these plots, revealing the teeth geometry has the same effect in both modes of operation of the gear unit.

<img 
 style="display: block; 
 margin-left: auto;
 margin-right: auto;
 width: 50%;
 text-align: center;"
 src="https://uptimai-website.s3.eu-central-1.amazonaws.com/scatter_fha_r_median_clean_eb93db4963.png" 
 alt="scatter_fha_r_median_clean">
</img>
Fig. 4. Comparison of noise levels with- and without the load of the gear

## Benefits

- **Reliable modelling despite the low number of measurements**. Models of the noise level of 16 input variables were built from only 50 data samples and achieved approx. 90% of overall reliability.
- **The main source of the noise** in the electronic brake **was identified**. The right focus on sources of noise was the first step to successful design. 
- **Design suggestions** were made **to reduce the noise level** generated by the braking system. Optimization of multiple outputs allowed to improve both modes of braking.

large_pebrake-control-crop.jpg

medium_pebrake-control-crop.jpg

small_pebrake-control-crop.jpg

thumbnail_pebrake-control-crop.jpg

pebrake-control-crop.jpg

Reducing the noise of the braking unit

Human performance in harsh environment
Monitoring workers under thermal stress and cognitive loading
ICARUS ARMOR is  an ESA SPARK funded project developing a product for an active monitoring of stress and fatigue of individuals operating in harsh environments, such as firefighters, field police, and heavy machinery operators.

## Challenge
The imposition of thermal stress on the human body can lead to a significant decrement in cognitive function, a critical factor when assessing the potential for accidents or serious incidents in occupational or extreme environments. Cognitive functions, including attention, memory, and decision-making, are particularly vulnerable to the effects of temperature extremes. These cognitive impairments can compromise safety, increasing the risk of errors and accidents in situations requiring high levels of attention and decision-making capabilities.

![icarus v2.png](https://uptimai-website.s3.eu-central-1.amazonaws.com/icarus_v2_7fe494024a.png)

*Fig 1. ICARUS ARMOR logo.*

The relationship between thermal stress and cognitive dysfunction underscores the importance of mitigating these stressors to maintain cognitive integrity and prevent serious incidents. Understanding the mechanisms by which thermal stress affects cognitive function is crucial for developing effective interventions and policies to enhance human performance and safety in extreme environments. The active monitoring of the current stress level imposed on a person is thus a key aspect of assessment for the current stress level of an individual so these policies can be applied in reality.  

## Solution
The person under monitoring, who is under extreme stress, shall not have the comfort to self-determine if they are under or over some stress limit. That is the purpose of beforehand determining their custom human digital twin model. A general model was created first during a test campaign that took place in the climate chamber at the Faculty of Electrical Engineering and Communication of the Brno University of Technology (BUT). The experiments were designed to evaluate the cognitive performance and physiological responses of human subjects under two distinct climatic conditions, during which each subject undertakes a battery of cognitive tests. Throughout the duration of the test, physiological responses of the human participants are captured utilizing lightweight wearable non-invasive measuring devices.

![Ales.png](https://uptimai-website.s3.eu-central-1.amazonaws.com/Alex_b7966139e1.png)

*Fig. 2. Aleš Svoboda (left) - a member of the ESA astronaut reserve, VIP participant of the testing campaign (photo BUT)*

Volunteers participating in the test included Aleš Svoboda, an army pilot and member of the ESA astronaut reserve. All the data obtained from wearables were stored into the database.   In principle the exact model of a wearable does not matter to be used in the system, which makes it open for a variety of users and use cases. Each type of measured biosignals were first processed and normalized by Uptimai for later use. Again, the process allows adding a type of measured biosignal if necessary. 
Based on the database, Uptimai prepared a set of mathematical models describing relations between physical responses of subject, environmental loading and cognitive loading. Building of these mathematical (or so-called surrogate) models takes only a few minutes of computational time so it is very easy to update them for either a group (e.g. workers of a certain profession) or an individual. 

![uptimai postprocessor.png](https://uptimai-website.s3.eu-central-1.amazonaws.com/uptimai_postprocessor_212e09c06d.png)

*Fig. 3. Analysis of surrogate model in the Uptimai software*

Also, models created within the ICARUS project were analyzed to identify the main physical responses related to environmental and cognitive loading. It was found that biosignals obtained using a breathing mask had only a minor response to cognitive loading in the test. Thus, it could be omitted from the set of wearables from the online monitoring because it is the most uncomfortable device for the user to wear and requires additional maintenance.

![Cognitive loading - workflow 2.png](https://uptimai-website.s3.eu-central-1.amazonaws.com/Cognitive_loading_workflow_2_3f0fd0e3ca.png)

*Fig. 4. Active monitoring of a person under thermal stress*

Fig. 4 shows how Uptimai software and its models are used during the active monitoring of a user under environmental and cognitive stress. The person under stress is sending data from their wearables to the database. Data is then processed by Uptimai scripts and fed to the previously built Uptimai surrogate model. The model producess the computed level of the current equivalent cognitive loading of the test subject. Based on the previous testing campaign it is possible to state if the person under monitoring is closing to their limit (threshold) of the overall cognitive load that was found during the testing phase. In such cases the system sends an alert message to the operator responsible for withdrawing the person from the current task in a hostile environment.

## Benefits
- **Versatile sets of hardware.** Monitoring can be enhanced with any type of wearable to respond to users and their custom needs.
- **Models can be easily updated.** Either for individuals or for professionals, building of mathematical models from biosignals is a matter of just minutes.
- **Safe and efficient deployment of workers.** System is warning people under active monitoring when their cognitive performance is compromised.

![logo esa tb_hvid_text.jpg](https://uptimai-website.s3.eu-central-1.amazonaws.com/logo_esa_tb_hvid_text_266cf2b0e2.jpg)


medium_icarus v2.png

small_icarus v2.png

thumbnail_icarus v2.png

icarus v2.png

Human performance in harsh environments

TEKREVOLUTION is a high-tech service and consulting company from Italy with huge experience in R&D activities. One of their main activities is to explore intelligent additive manufacturing to provide higher stress-to-weight ratio solutions for the market.

## Challenge
For the space sector, it is important to reduce the weight to the minimum possible, while maintaining excellent performance. That is why it is very important to choose the right selection of materials for each component of the structure, depending on the stress supported and its density.

In this case, we are analyzing the selection of the optimum material for a retainer belonging to the Spring Separation System of Interstage 2/3 of the Vega C launcher (Fig. 1). The actual design of the retainer is made of Aluminium AL7050 and the component is manufactured by machining it from an Aluminium billet. Nevertheless, due that the load supported by the retainer is not very big (around 600 N), it is possible to use polymeric materials such as PPS CF, PEEK or Carbon PEEK using Additive Manufacturing techniques making the component lighter.

For this study, two main properties of the material are considered. The density, and the Young Modulus. The objective is to see in which ranges those variables can be while matching the structural behavior in terms of maximum displacement, stress supported and natural modes to understand which kind of polymeric material is more efficient for this task. It is important to remark that some constraints on the inputs and outputs were set during the optimization process to avoid unrealistic materials, and that a Safety Factor = 2 was requested in terms of allowable Von Misses stress. 

![retainer_image4.jpg](https://uptimai-website.s3.eu-central-1.amazonaws.com/retainer_image4_016d92b219.jpg)
*Fig 1. Retainer geometry*

## Tools
For this problem, we used two Uptimai new optimization features. The first one is the Direct Optimizer which aims at minimizing (or maximizing) a function of interest with one single output with so-called Evolutionary Algorithms which mimic the behavior of a population of living beings in an environment subject to natural selection. Inside the Direct Optimizer we are proposing two different algorithms, depending on the problem: Differential Evolution (DE). This algorithm creates a population of potential samples and makes it evolve at each generation. The evolution process involves independent and random mutation of each coordinate of each member of the population. This evolution process is repeated multiple times in a looping fashion and at each iteration, the population members are replaced by the new members (children) only if they provide a better result (lower function value).

The algorithm stops when the population has converged closely enough to a minimum point or the maximum allowed number of function evaluations is attained. Because each coordinate is mutated independently this algorithm is subject to the curse of dimensionality. This is why the decoupling option is recommended for high dimensional problems (dimension >= 10). Hybrid method of DE and Covariance Matrix Adaptation. The Covariance Matrix Adaptation (CMA) is an algorithm that evolves the population by generating children according to a multivariate Gaussian distribution. The best child elements are selected for the next population and the parameters of the Gaussian distribution are adjusted in a heuristic manner. Again the algorithm stops in case of convergence of the population or if the maximum number of function evaluations has been reached. The Hybrid Method performs a decoupling process to separate 1D-subproblems and ND-subproblems. Then, the DE algorithm is applied to 1D-subproblems and the CMA algorithm is applied to ND-subproblems.

To tackle the curse of dimensionality, for both algorithms, the UptimAI software use a decoupling process (also known as variable separation in the literature) that transforms the main domain into smaller subdomains. Then the optimization process is separated into lower-dimensional subproblems, providing better performance and converging faster.

On the other hand, the Multi-Objective Optimizer is designed to minimize multiple concurrent outputs. It works in a similar way that the Direct Optimizer does. However, it is not usable with decoupling and uses an evolutionary algorithm called NSGA-II. Unlike the Direct Optimizer, the Multi-Objective Optimizer provides multiple solutions approximating a Pareto Front of the problem.

## Solution
We coupled the UptimAI tool with the simulation software MSC Nastran, without the need of interfering with its code or interface. Then the MSC Nastran started running simulations for the sample points (different material properties) indicated by the UptimAI algorithm for converging into the optimum solution.

![retainer_image1.png](https://uptimai-website.s3.eu-central-1.amazonaws.com/retainer_image1_7d8e8dd2f4.png)
*Fig 2. Results obtained from Nastran MSC for a specific sample*

For this case, approximately 500 simulations were performed to get a converged solution. Uptimai Multidisciplinary Optimization software gave as result the Pareto front of the duo density - young modulus that was improving both displacement, Natural mode and stress (the last one isn’t represented in Fig. 3). From the results obtained, we could extract the properties of the wanted material and by consequence the type of polymeric material that should be used to reduce the weight of the retainer while maintaining an excellent performance.

![retainer_image3.png](https://uptimai-website.s3.eu-central-1.amazonaws.com/retainer_image3_e78fdb3425.png)
*Fig 3. Pareto Front showing zone of optimum properties, and the outputs achieved*

## Benefits
- **Interpretation of variables effect.** Understanding the effect of each variable on the output
- **Optimization of the results.** Obtention of the Pareto Front that is improving all the outputs at the same time
- **Reduction of computational cost.** Our increased convergence rate makes that fewer samples are needed to achieve the solution.

Design, analysis and optimization of retainer materials

medium_tekrev.png

small_tekrev.png

thumbnail_tekrev.png

tekrev.png

small_retainer_image2.png

thumbnail_retainer_image2.png

retainer_image2.png

Retainer Optimization

IGW POWER drives solutions for gears, gearbox assemblies and prismatic parts. Although most known for its efficient and reliable products for the rail industry, the company also supplies a diverse amount of applications and markets from construction machinery to medical devices.

## Challenge
Besides the overall reliability and durability, the noise characteristics are one of the key topics for the gear and gearbox design. This parameter is expected to be as low as possible since it strongly affects the operability of the final product and its environment.

The current state of the art in gear noise analysis is the finite element method approach. For the initial stages of the design, it is important to quickly identify parameters of the design crucial for its noise characteristics. However, the cost of such an activity cannot be excessive from the perspective of time and resources. This is why the fast Uptimai Preliminary analysis tool was used for the first sensitivity analysis of the gearwheel.

The goal of this preliminary study was to evaluate the sensitivity of noise-related phenomena to 6 geometrical parameters of the gearwheel. Using the method of Uptimai, only 19 computations were required to reach this goal, which can be later used as a baseline for follow-up design steps.

![gearwheel.png](https://uptimai-website.s3.eu-central-1.amazonaws.com/gearwheel_eba402edf0.png)
*Fig. 1. Example of the analysed gearwheel type*

## Solution
This type of very fast analysis was also used because there was no fully automatic connection between the wheel geometry model of the wheel, the meshing software and the MSC Nastran that provided the amplitude-frequency characteristics of the gearwheel noise. This increased the cost of each simulation and therefore only very few were available to be made. The main focus of the study was on the maximum noise level (equivalent radiated power, ERP), frequency of this ERP peak, and the probability of resonating frequencies to occur.

![gearwheel_image2.png](https://uptimai-website.s3.eu-central-1.amazonaws.com/gearwheel_image2_b3836f4e78.png)
*Fig. 2. Example of the extracted amplitude-frequency characteristics*

During this process, it is the Uptimai algorithm that decides which input parameter combinations have to be computed in the next step. These input data are provided in small batches, where every new iteration is proposed based on the results of the previous ones. This adaptivity gives the analysis the key advantage in accuracy and cost over traditional methods of statistical analysis.

![gearwheel_image3.png](https://uptimai-website.s3.eu-central-1.amazonaws.com/gearwheel_image3_7d179739fd.png)
*Fig. 3. Sensitivity analysis of the equivalent radiated power and its frequency*

As a result, the Uptimai sensitivity analysis was presented in the post-processing environment, showing the effect of wheel geometry parameters on the overall variance of results. For three of four observed output variables, parameters defining the gearwheel rib thickness (especially A1) are among those most influential. E.g. for the frequency with the highest ERP level (output 2), the effect of other parameters was found to be negligible.

![gearwheel_image1.png](https://uptimai-website.s3.eu-central-1.amazonaws.com/gearwheel_image1_43453fb14a.png)
*Fig. 4. Sensitivity analysis of output related to resonating frequencies*

However, the distance of holes in the rib from the axis and their number affects the maximum of ERP as well. The number of holes in the rib, R9, becomes partially important when finding if the last observed ERP peak frequency is in the resonance with the previous one, the number of holes is the key input parameter. (see Fig.1 for reference).

The only parameter neglected for all outputs is the hole diameter D_6. Other input parameters related to holes in the rib are probably relatively much more significant than this one and D_6 is just “hidden in their shadow”.

This, and other findings as well, can be gathered within the following analysis steps. A simple model will be created from additionally computed samples, allowing to find the effect of interactions between input parameters. Then, the creation of the full statistical model reveals the complete set of insights into the design.

## Benefits
- **Definition of the problem.** Clear identification of relevant variables.
- **Efficient process.** A fast algorithm requires a minimum number of samples.
- **Insights for future improvement and optimization of the gearbox.**

small_gearwheel_image4.png

thumbnail_gearwheel_image4.png

gearwheel_image4.png

Gearwheel sensitivity analysis

ŠKODA AUTO is the largest car manufacturer in the Czech Republic and part of the Volkswagen Group since 1991. The company uses highly advanced simulation tools in its R&D departments to meet all design goals, where passenger safety is one of the top priorities.

## Challenge
Although side collisions occur less likely than front or rear crashes, it is still a very common type of car accident, especially in cities. Moreover, at the same speed, these can cause more severe injuries to passengers inside the vehicle. Thus, it is crucial for design engineers to know the most important aspects of side crash behavior. Finite element method computations are used to analyze the impact of a car collision on passengers. Computations are performed repeatedly with variations in design parameters to see their effect on forces acting on passengers. However, the total number of FEM computations is limited since these can be a lengthy process even with powerful computers. Side crash is a complex problem depending on a large number of parameters and it was not possible to investigate all of them in this study. In the first step, the ŠKODA AUTO experts simplified the problem and selected 11 parameters which should be important or interesting to analyse. In the second step, a preliminary analysis applied on these 11 preselected parameters to assess their importance. Then within just 65 computations, the preliminary analysis of Uptimai software was able to propose a reduction of this list further to 6 most influential parameters.

![side_impact_image4.jpg](https://uptimai-website.s3.eu-central-1.amazonaws.com/side_impact_image4_1261be2614.jpg)
*Fig. 1. Side collision testing, photo by Euro NCAP*

## Solution
In the beginning, we coupled the Uptimai tool with the simulation software PAM-Crash using the sampling based approach. Then, the UptimAI tool began to set combinations of inputs for the PAM-Crash, automatically varying design parameters of the car. The **Uptimai algorithm** starts to analyze the task using a very effective logical scheme. From the results of only 40 FEM simulations, the program is able to identify the sensitivities of input variables. This is about 50% of simulations required by the second-order polynomial chaos, an advanced statistical method used today.

The **Uptimai sensitivity analysis** allows to see not only the effect of inputs on the overall variance of the results but also how they affect the mean value of the probabilistic result from the so-called preliminary surrogate model. As mentioned earlier, the main goal of the analysis so far was to perform the sensitivity analysis with the lowest possible use of resources. To complement data with FEM results required to build the preliminary surrogate model, additional PAM-Crash calls were made. In the end, 65 samples were used in total for both observed outputs - rib force and rib deformation. This is a significantly lower number than for design-of-experiment methods commonly used in engineering. E.g. Box-Behnken experimental design would require almost three times more FEM computations for this task with 11 input variables.

![side_impact_image2.png](https://uptimai-website.s3.eu-central-1.amazonaws.com/side_impact_image2_d484be6a61.png)
*Fig. 2. Sensitivity analysis of rib force and rib deformation*

The information learned from the sensitivity analysis was further visualized in the **Uptimai histograms**, showing clearly the effect of each input variable on the probability distribution of results. The so-called influencer plot is the graphic representation of changes in results caused by the selected parameter. Thus, engineers had a straightforward illustration of changes in the mean value and the variance of results.

![side_impact_image1.png](https://uptimai-website.s3.eu-central-1.amazonaws.com/side_impact_image1_a16e6ed302.png)
*Fig. 3. Example of an input parameter influencing only one output*

Another piece of information gained from the model was the probability of inputs to interact with the others. It was only possible due to the novel methodology used in the Uptimai software, its smart scheme and principles of model residuals handling. Based on all these data, the number of input parameters was reduced from 11 to 6. In the following stage of the project, this allowed to obtain the detailed surrogate models of both outputs for only 255 FEM simulations. The model of acting forces was the more demanding, the model of deformations used 99 simulations and these were already solved for the first output.

## Benefits
- **The low number of required FEM simulations.** Reducing the total number of simulations to 50%.
- **Identification of influential parameters.** Fast recognition of variables with high impact on results.
- **Visualisation of effects of variables.** Better handling of analysis results with illustrative plots.

small_side_impact_image3.png

thumbnail_side_impact_image3.png

side_impact_image3.png

Side Impact Analysis

Turbo-fans are one type of airbreathing jet engine that is widely used in aircraft propulsion. Despite being more complex, it can obtain higher efficiencies than its brother the turbojet for subsonic flights.

## Challenge
Designers of turbofans have to increase the Force that is able to produce together with the Specific Impulse, which is a measure of the efficiency of the engine. The result of this is normally an engineering trade-off as both outputs are independent.

![turbo_optimization_image4.jpg](https://uptimai-website.s3.eu-central-1.amazonaws.com/turbo_optimization_image4_b5ce3689a1.jpg)

The objective is to study the performance of multiple staged compressors, and to preliminary design the pressure ratio desired for every stage, taking into account the physical uncertainties of flight conditions.

For this problem, we were studying how seven parameters were affecting both the Thrust Force and the Specific Impulse to get insights into how each variable is affecting the solution and how it is possible to optimize both outputs at the same time. It was a complex problem, as the inputs were interconnected, and multiple combinations of them were left for unfeasible options.

## Solution
We started the project by coupling the Uptimai tool with the calculation software. In this case, the external software was a Python script, that was reading the inputs given by Uptimai software, computing all the outputs and feeding its result back. With these results, the AI would choose where would be more efficient to locate the following sample adaptively, and automatically repeat the process until the desired accuracy is achieved.

![turbo_optimization_image6.jpg](https://uptimai-website.s3.eu-central-1.amazonaws.com/turbo_optimization_image6_609697a2c7.jpg)

Our **Uptimai Sensitivity Analysis** showed us which were the most relevant parameters and which ones could be neglected as its effect was almost negligible. For both outputs, we saw that the inputs in which we should focus to improve the design were in the compressor side, in particular the pressure ratio that both the low and high compressors are able to produce. In the case of the Adimensional Force, the bypass ratio was also a very important factor.

![turbo_optimization_image3.png](https://uptimai-website.s3.eu-central-1.amazonaws.com/turbo_optimization_image3_649c25c43e.png)

To understand how those relevant variables are actually affecting the outputs we used the **Uptimai Increment Functions**. That allows us for both, for learning about how each variable was affecting each output, but also as a preliminary optimization tool, where we could distinguish in which ranges each variable should be to maximize the output. In the case of the bypass ratio, we found that the best option for improving the Thrust Force would be to have it between 10 and 13.

![turbo_optimization_image5.png](https://uptimai-website.s3.eu-central-1.amazonaws.com/turbo_optimization_image5_4d21fd2a85.png)

Thanks to the **Uptimai Histograms** we could get a statistical view of how the turbofan was behaving, and the uncertainty around it. As can be seen, there are some combinations of inputs that make the engine behave in a non-desirable way (below Fad = 3), but thanks to the influences we could tune the parameters to be always above 4.5.

![turbo_optimization_image2.png](https://uptimai-website.s3.eu-central-1.amazonaws.com/turbo_optimization_image2_7267903a50.png)

As with the studies with the meta-model, we saw that it was impossible to get a particular design point that was maximizing both outputs (some variables had conflictive effects), we used our **MultiObjective Optimization** feature. With the MOO, we were able to get the Pareto Front that was maximizing both outputs, with the selected constraints for the outputs. That allowed to choose effectively the design point for different engine needs.

## Benefits
- **Definition of the problem.** A clear description of relevant variables
- **A deeper understanding** of the physics behind the problem, and how each variable affects the outputs
- **Multiobjective optimization.** Having all the possible values that are maximizing at the same time both outputs


Optimizing Force and Specific Impulse of a turbofan

medium_turbo_optimization_image1.png

small_turbo_optimization_image1.png

thumbnail_turbo_optimization_image1.png

turbo_optimization_image1.png

Turbo Engine Preliminary Optimization

VARROC LIGHTING SYSTEMS is a leading global passenger car exterior lighting supplier. It provides complete solutions from design to manufacturing, which are supported by engineers and other experts in a worldwide network of modern R&D centers and factories.

## Challenge
Modern automobile headlamps are complex optical components consisting of a large number of parts. The optics inside do not only work with the light shining out from the headlamp but also reflect sun rays coming from the outer world. In some cases, these might cause local overheating of certain areas inside the headlamp and lead to damage of used plastic materials. The presented study is a typical task of the optical department. Its purpose is to identify the sun direction leading to the highest concentration of sun rays on each part of the headlamp. The sun direction is defined by its azimuth and inclination, the sunlight intensity was set to a constant equal to the possible maximum. Thus, the results are always on the safe side regardless of the daytime or the orientation of the vehicle towards the Sun. The resulting maps of the sunload, created for each headlamp part, are an important input for further examinations using FEM computations studying the ability of a given sunload to create critical temperatures in headlamp parts.

## Solution
The first step of the project was the coupling of the Uptimai tool with Varroc’s internal software for ray-tracing simulations. These two software use open-format files for communication, thus, there was no need to interfere with their codes. Within the created automatic loop, the **Uptimai algorithm** was preparing inputs for the ray-tracing software which supplied the algorithm with sunload results. Every other set of sunload computations was set to address sun directions leading to a significant response in the sunload of the examined headlamp part. There is no fixed grid of azimuth and inclination angles as in the standard process of sunload analysis. Contrary, the algorithm adaptively reacts to variations of results received from the ray-tracing software, actively predicting sun direction angles with the high possibility of an excessive value of the sunload. The step between sun direction angles is set automatically by the algorithm according to the actual needs of the solved problem without other interventions of the user. Thus, there is an optimal precision-to-cost ratio of the computation process. The resulting maps of the sunload can be depicted in the **Uptimai function plot** postprocessing tool. From the interpolation of the prepared surrogate model of the sunload, it can obtain results for any combination of azimuth and inclination angles within the examined domain. It also supports results generation based on a list of prescribed input combinations, so the user is able to prepare the usual analysis outputs presented on an even grid with the specified resolution.

![sunload_image2.png](https://uptimai-website.s3.eu-central-1.amazonaws.com/sunload_image2_351ab80894.png)

*Fig. 1. Resulting sunload maps generated for selected headlamp parts*

## Benefits
- **Quick identification of peaks in sunload.** The algorithm is capable of finding all important local extremities of sunload.
- **Decreasing the computational time.** Adaptive seeking for variances in the sunload is much more efficient than the analysis on a fixed grid.
- **Comprehensive result review.** From the surrogate model, it is possible to generate sunload maps on various grid resolutions.

Speeding up the ray-tracing inside the headlamp

medium_sunload_image1.png

small_sunload_image1.png

thumbnail_sunload_image1.png

sunload_image1.png

Sunload Simulation Process

## Optimization of fan characteristics under various flow-rate
CFD SUPPORT is an engineering company dealing primarily with the CFD and FEA simulations, and also other activities related to numerical simulations of physical phenomena. Its own-developed TCAE software is a comprehensive environment of standalone modules for engineering simulations.

## Challenge
Large centrifugal fans may be used for various industrial applications, such as in ventilation units for manufacturing facilities. Although the fan presented in this study is not a part of a used solution, it was derived from an existing fan, for which the comparison of the CFD results with measurement has been made. The study shows a comprehensive analysis of main fan characteristics. These were observed for the varying volumetric flow rate, which is a typical flow parameter for analysing the fan performance. Additionally, the impeller geometry has changed during the analysis to find the most promising fan configuration for a given range of volumetric flow rate and a fixed volute geometry.

![centrifugal_fan_image2.png](https://uptimai-website.s3.eu-central-1.amazonaws.com/centrifugal_fan_image2_6de673d6b9.png)
*Fig. 1. Centrifugal Fan in the volute*

Six parameters (volumetric flow rate, blade radius, blade thickness, number of blades, blade angle, b1,2 – blade depth), and six outputs (efficiency, total pressure difference, power, flow number, compression number and axial force) had to be evaluated in the study. During the analysis, the Uptimai tool successfully dealt with discontinuities in results, localizing them properly while suppressing oscillations in results. The created surrogate model can be used for the prediction of centrifugal fan characteristics without additional CFD computations.

## Solution
At the beginning of the project, the Uptimai tool was very easily connected to the automated CFD analysis process used by the CFD Support company in their TCAE simulation environment. This loop received values of input parameters from the Uptimai software and was able to automatically generate the geometry of the fan, create the computational mesh, and run and post-process the CFD simulation. Values of input parameters were ordered by Uptimai’s smart algorithm and then it used CFD results to build surrogate models of the fan’s characteristics.

![centrifugal_fan_image5.png](https://uptimai-website.s3.eu-central-1.amazonaws.com/centrifugal_fan_image5_0657fcb6a4.png)
*Fig. 2. Simulated flow through the fan*

The **Uptimai Sensitivity Analysis** (Fig. 3.) confirmed the importance of the volumetric flow rate through the fan as the main flow parameter, which has to be considered in the fan design. From the impeller parameters point of view, the width of the impeller (b12) and the number of its blades have a main effect on the overall performance. Partially, it is due to the fact these inputs are interacting with the volumetric flow rate – the best setting of geometric variables depends on the flow rate. Thus, these interactions need to be considered when designing the fan to be working well in a wider range of the flow rate.

![centrifugal_fan_image1.png](https://uptimai-website.s3.eu-central-1.amazonaws.com/centrifugal_fan_image1_91a01e8d3f.png)
*Fig. 3. Sensitivity Analysis of the Total Pressure Difference*

Then, in the **Uptimai Increment plot** (Fig. 4.) is shown exactly how the efficiency of the fan is dependent on the flow rate and the impeller width. As indicated by the zero efficiency, it can be noted that the impeller of very low width is incapable of handling high flow rates. It suggests the width b12 should be higher than 0.2m to ensure there will be a reasonable efficiency of the fan for the whole range of possible flow rates. Also, it is the way to avoid excessive axial forces which might lead to an eventual fan malfunction.

![centrifugal_fan_image4.png](https://uptimai-website.s3.eu-central-1.amazonaws.com/centrifugal_fan_image4_8ebcc6300d.png)
*Fig. 4. Fan efficiency (left) and axial force (right) based on impeller width and volumetric flow rate*

The **Uptimai Histogram** Plot (Fig. 5.) shows the actual potential of the design under various operational conditions. There is a high probability to reach an efficiency of about 80% with a maximum of 86%. On the other hand, the histogram confirms the possibility of designs not matching specific flow rate conditions. As demonstrated above, these can be avoided by restricting ranges of geometry parameters.

![centrifugal_fan_image6.png](https://uptimai-website.s3.eu-central-1.amazonaws.com/centrifugal_fan_image6_f87d668781.png)
*Fig. 5. Probability distribution of efficiency of all theoretically possible designs*

New ranges of inputs were identified using the **Uptimai Preliminary Optimization** tool which proposes input distributions leading to the statistical improvement in results. Once the volumetric flow rate was one of the input parameters, the optimization tool could be also used to visualize the flow rates best fitting for the current design concept. As seen in Fig. 6., it was generated for all outputs and these were compared against each other. It showed the best efficiency for flow rates about 25m/s and was checked if there is a conflict with e.g. axial forces.

![centrifugal_fan_image7.png](https://uptimai-website.s3.eu-central-1.amazonaws.com/centrifugal_fan_image7_a94dd0a583.png)
*Fig. 6. A trade-off can be easily done by comparing of Regions of Preference/Avoidance generated for different outputs*

## Benefits
- **Definition of influential parameters.** There is only a minor effect of selected impeller parameters other than its width and also the number of blades.
- **Comprehensive design review.** Taking multi-objective considerations into account for optimization of different outputs with specific constraints.
- **Identification of fan usability.** Investigation of the design limits of all the variables to assure a correct fan operability.

thumbnail_cfdsupport.png

cfdsupport.png

small_centrifugal_fan_image3.png

thumbnail_centrifugal_fan_image3.png

centrifugal_fan_image3.png

Centrifugal Fan

The VIRTUAL VEHICLE Research GmbH is Europe’s largest R&D center for virtual vehicle technology, the innovation catalyst for future vehicle technologies. Their focus is on the linking of numerical simulations and hardware testing, creating automated testing and validation procedures.

## Challenge
The fastly growing e-cars market excites the need for advanced methods for design and analysis specifically for this sector. VIRTUAL VEHICLE works on modelling the vehicle battery electric behavior, allowing improvements in the battery design. For the “Single-Particle” model (SPM), an optimal setting for the model’s 31 parameters was found in the previous research to obtain the best-known match with the measured data.

In this study, the goal was to enrich existing data with comprehensive statistical insights. In the initial stage, the problem was split into a series of iteratively created surrogates. First iterations were dedicated to searching for domain limits and identification of parameter ranges for a stable battery model. Then, the domain was restricted to the effect on statistical aspects of results from iteration to iteration.

Another part of the project investigated the close surroundings of the optimum used as the reference. The statistical analysis examined the robustness of the solution and gave the answers to whether or not some parameters may be omitted from the model. Also, it revealed there is still room for fine tuning of parameters to achieve the match between modelled and measured battery cycle.

![battery_cycle_image3.jpg](https://uptimai-website.s3.eu-central-1.amazonaws.com/battery_cycle_image3_bedc5ee3d8.jpg)
*Fig. 1. Electric vehicle battery (photo by Gereon Meyer CC BY-SA 4.0)*

## Solution
Coupling of the Uptimai tool with VIRTUAL VEHICLE’s FEMToolbox software for battery cycle simulation was easily done without the need of interfering with neither simulation nor UQ code. The already existing interface of the VIRTUAL VEHICLE’s software was modified to pass data to the Uptimai tool after post-processing of the simulation results. Uptimai algorithm used these outputs created for altering input parameters to build a surrogate model of the behaviour of the simulated battery cycle. Results presented here are for the model created for the close surroundings (20% of the complete input range) of the reference point.

The **Uptimai algorithm** was used to process simulation results into the metamodel of agreement between measured and computed battery-cell cycles. The level of agreement was defined as normalized Root Mean Square Error (RMSE), observed for each simulation. The metamodel consists of dependencies between input parameter changes and the simulation results, allowing a detailed description of the battery cycle.

Specific response to changes in particular input variables was observed using the **Uptimai Increment plot**. This feature allows treating each dependency separately but is also able to perform a cross-comparison of variables against each other. Fig. 2 depicts the total response to the three most influential variables. Note the peak in results caused by the cumulative effect of these variables close to their upper boundary. This combination of input parameter values should be avoided when seeking a better battery cycle model.

![battery_cycle_image4.png](https://uptimai-website.s3.eu-central-1.amazonaws.com/battery_cycle_image4_52076fb2fd.png)
*Fig. 2. Extremity localized on the increment function plot*

The **Uptimai Sensitivity Analysis** confirmed the tight connection of variables through high-order interactions, which in sum create a considerable portion of the uncertainty of results. For the most influential variable ocv_Li(ci)_1, interactions are responsible for 2/3 of variance in results caused by this input variable. Although the surrogate model was much simpler than in the above-mentioned case of the iterative approach, mutual interactions of up to four variables had to be investigated.

![battery_cycle_image5.png](https://uptimai-website.s3.eu-central-1.amazonaws.com/battery_cycle_image5_c7bc6c1a0d.png)
*Fig. 3. Sensitivity Analysis of the reduced domain*

Also, another surrogate model was created for seven input variables indicated as statistically negligible in all previously done UQ analyses. It confirmed that even in this highly focused domain some variables, especially D_1, are not able to affect the results (see Fig. 3). As proven by the **Uptimai Histogram Plots** (Fig. 5), the cumulative response of all variables in the reduced domain is much lower than the effect of the most influential input. However, since the nominal point of the reduced domain was set to the so far known optimum, findings of this very precise surrogate were used to obtain an even better result. The new optimum increases the quality of the battery cycle model by more than 20%.

![battery_cycle_image2.png](https://uptimai-website.s3.eu-central-1.amazonaws.com/battery_cycle_image2_70a71e5e2f.png)
*Fig. 4. Comparison of effects of the most influential variable of the full domain against all variables of the reduced domain*

## Benefits
- **Definition of valid design space.** The Uncertainty Quantification was able to explore and identify correct ranges for the input domain where the Single-Particle model always converges.
- **Suggestions for stable optimum** – addressing ranges of input parameters responsible for better-than-average results allowed to **improve the current optimum by 20%.**
- **Deep statistical insights** into the Single-Particle model behaviour with **identification of the most influential input variables.**

Electro-chemical battery model validation

medium_battery_cycle_image1.png

small_battery_cycle_image1.png

thumbnail_battery_cycle_image1.png

battery_cycle_image1.png

Discharge battery cycle

## Design, analysis and optimization of a CubeSat structure
TEKREVOLUTION is a high-tech service and consulting company from Italy with huge experience in R&D activities. One of their main activities is to explore intelligent additive manufacturing to provide higher stress-to-weight ratio solutions for the market.

## Challenge
For the space sector, it is important to reduce the weight to the minimum possible, while maintaining excellent performance. That is why it is important to comprehend that it is more optimum to have different thicknesses and material properties through different zones with particular characteristics. Moreover, it is interesting to understand which parts of the component are affecting more to the behavior when loaded. For this reason, in this case, the component was divided into 10 different parts whose properties would be studied individually. Three different aspects were examined, the displacement of the center of mass of the payload, the first natural mode and the maximum stress supported with the objective of minimizing the CubeSat weight. Overall, we considered the thickness of each of the 10 parts that composed the model and the Young Modulus of the material. A high resolution study was made with only 143 Nastran simulations.

![cubesat_image5.png](https://uptimai-website.s3.eu-central-1.amazonaws.com/cubesat_image5_e3f9af6106.png)
*Fig. 1. Scheme of the CubeSat Structure*

## Solution
We coupled the UptimAI tool with the simulation software Nastran, without the need of interfering with its code or interface. Then the Nastran started running simulations for the sample points indicated by the UptimAI algorithm for creating a metamodel of the behavior of the component under load.

The **UptimAI Sensitivity Analysis** (Fig. 2.) helped to identify which parts of the component were the most crucial, and supported most of the load, measuring which input had more influence in the three outputs (displacement, first natural mode and stress). Also, it was very useful to determine which parts of the component weren’t supporting the load at all, so its thickness could be reduced without affecting the overall performance at all. In this case, only the thickness of the load area had an important effect on the maximum stress supported by the structure, being able from now on to focus on that two parts and knowing that it is possible to reduce the thickness on the other parts to reduce the weight without affecting the stress performance.

![cubesat_image4.png](https://uptimai-website.s3.eu-central-1.amazonaws.com/cubesat_image4_daed8090cc.png)
*Fig. 2. Sensitivity Analysis of the Maximum Stress on the CubeSat*

Details of specific zones of the domain can be easily seen in the **UptimAI Increment plot** (Fig. 3.). It shows the shape of increment functions illustrating the interactions and dependencies between the values of the input parameters and the displacement. That allows to easily identify high response zones, for avoiding in the design phase and reduce both the variability of the displacement and to reduce its average response. In this case it was necessary to have at least one of the loading areas with a thickness higher than 2.5 mm, to operate in a stable condition.

![cubesat_image3.png](https://uptimai-website.s3.eu-central-1.amazonaws.com/cubesat_image3_bc67d9500b.png)
*Fig. 3. Interaction between thicknesses of loading parts in the displacement of the payload*

The histogram (Fig. 4.) is showing the position of the 1st Natural Mode. The Nastran computations were cut to 120 Hz as maximum because there is no interest in knowing the exact value of the first frequency if it is higher than this value, as it already surpasses the safety restraints (NOTE: the metamodel accurately caught this aspect). The histogram allows to check which is the average response (around 110 Hz), and to study in detail the input conditions of the left tail of the distribution that make appear undesired behavior of the structure.

![cubesat_image6.png](https://uptimai-website.s3.eu-central-1.amazonaws.com/cubesat_image6_6f8607e641.png)
*Fig. 4. Statistical Modal Response of the CubeSat*

Finally, the engineers used the **UptimAI Preliminary Optimization** tool to establish new input distributions. These proposed distributions show which zones of the domain carry an statistical improvement, and which zones should be avoided. That way it helps the engineering team on the decision making of which combination of input parameters are more interesting to explore. Using the thickness distributions for the loading parts suggested by the UptimAI algorithm, the average displacement was halved and the variance was largely decreased, leading to a better and more reliable performance.

![cubesat_image1.png](https://uptimai-website.s3.eu-central-1.amazonaws.com/cubesat_image1_343b1974f0.png)
*Fig. 5. Statistical response of the displacement of the payload before and after improvements*

## Benefits
- **Definition of influential parameters.** The parameters of the component that were relevant to the results were reduced by more than 60%
- **Optimization of the results.** Decreasing the average displacement of the payload to the half and its variance to input parameters while complying with modal restrictions.
- **Deeper insights** into the structural behavior of the component under high loadings.


Reduction of weight while increasing performance

medium_cubesat_image2.png

small_cubesat_image2.png

thumbnail_cubesat_image2.png

cubesat_image2.png

CubeSat structural design

IDIADA CZ is part of an international engineering company Applus IDIADA. In the Czech Republic, more than 200 experts deliver a portfolio of services from technical simulations and design proposals over conception designs up to design and series manufacture for automotive, aerospace, and other industries.

## Challenge
In order to ensure good agreement between simulation and real part test, a robust material model must be available. Typically a tensile and creep sample tests are provided for calibration of the material model. Calibration of a material model is an iterative process of tuning several (typically around 10) parameters. After running the tensile and creep simulation using these parameters, a comparison to the original test data is made. Based on the correlation between simulation and the test data a new set of parameters is suggested.

Using standard simulation tools is time-costly proportionally to the number of parameters. Therefore it is desirable to decrease the number of iterations. This can be achieved by a smart analytical tool that can quickly recognize the optimal value of parameters.

This is what UptimAI’s solution excels at. Suggested algorithm decreased the number of iterations from thousands to several hundred. Furthermore, UptimAI has provided deep insight into parameter interaction.

For this study, the FEM structural solver Abaqus was used, allowing only a limited number of simulations within the timeframe of the project. Thus, the problem was described in a series of iteratively created surrogate models. In each iteration, a smaller section of the design space was selected in order to focus on ranges of parameters with a higher probability of a better match of material characteristics. Despite the surrogate model being built four times in total, the number of required computations dropped from thousands to hundreds.

![material_model_tuning_image6.png](https://uptimai-website.s3.eu-central-1.amazonaws.com/material_model_tuning_image6_173b5eadfb.png)
*Sample of the material under the virtual “testing” in Abaqus (picture: IDIADA CZ)*

## Solution
Coupling of Abaqus and UptimAI software was easy and straightforward. A standard approach was used, where input files required by Abaqus solver (these are text files in essence) were altered by UptimAI tool. After the simulation has finished, a unified post-processing python script produces resulting tensile and creep curves (in a form of a CSV table).

The **UptimAI algorithm** is used to process simulation results into the metamodel of agreement between measured and computed material characteristics. The level of agreement was defined as Normalised Mean Square Error (NMSE), observed separately for each material test (tensile test and three tests of creep – at stress 4MPa, 5MPa, and 6MPa). Results presented here are for the creep test at 4MPa.

The analysis of the metamodels began with the **UptimAI sensitivity analysis**. It revealed that in the first metamodel (the first iteration of the study) one parameter of the material model is dominant. Thus, the range of this parameter, C10, was reduced for the next iteration. There, other parameters showed their importance too, since local extremities they caused were more focused on now. Nevertheless, the effect of some parameters, like δς or δε, is still too negligible even for the last iteration created for the smallest input domain.

![material_model_tuning_image3.png](https://uptimai-website.s3.eu-central-1.amazonaws.com/material_model_tuning_image3_7929642653.png)
*Sensitivity analysis – on a smaller domain, the mathematical model is able to describe the problem in more detail*

The overall statistical properties of each metamodel were apparent from **UptimAI histograms**. It was found that the material model has a large overall variance, where any slight modification of the input parameters leads to significant changes in the accuracy of material characteristics. From the second iteration, it suggested the appearance of a distinct local extreme value. It also showed that interactions with parameters Sratio and n are responsible for this behaviour.

![material_model_tuning_image2.png](https://uptimai-website.s3.eu-central-1.amazonaws.com/material_model_tuning_image2_6b55f49798.png)
*Contribution of the parameter Sratio to the statistical result*

Position and origin of the extremity were later found on **UptimAI total incremental plots**. These depicted the sum of direct responses in the match of material curves to changes in particular input parameters and their combination. The figure allowed localizing the extremity with respect to ranges of material parameters, setting the base for the next iteration and reduction of the input domain.

![material_model_tuning_image4.png](https://uptimai-website.s3.eu-central-1.amazonaws.com/material_model_tuning_image4_51bf5da792.png)
*Total incremental plot – sum of increments to the output based on changes in three interacting input parameters*

New ranges of input parameters were set according to results of the **UptimAI preliminary optimization** tool. It used the statistical nature of the model to give recommendations leading to a new reduced input domain with a higher probability of a good match of material characteristics. This method also allows easy graphical comparison of proposed regions for multiple outputs. Here it has shown that model parameters affect all examined cases of creep behaviour (at 4MPa to 6MPa) in the same way. On the other hand, they are often contrary to the tensile behaviour, and thus, the iterative process was repeated again with the main focus on the tensile behaviour of the material.

![material_model_tuning_image1.png](https://uptimai-website.s3.eu-central-1.amazonaws.com/material_model_tuning_image1_fd23b7a519.png)
*Left: Example of proposed ranges of parameter Sratio inducing shift of output towards higher/lower values. Right: Adjusted probability distributions of NMSE values after all proposed input regions are applied.*

## Benefits
- **Definition of influential parameters.** The number of parameters could be reduced, since some of the inputs are negligible for the agreement of material characteristics.
- **Lower computational time.** Even though the process was iterative, the number of simulations required in total was cut significantly.
- **Deeper insights into the task.** Evaluation of the iterative process revealed the complex behaviour of the material model on a wide range of its input parameters.


small_material_model_tuning_image5.png

thumbnail_material_model_tuning_image5.png

material_model_tuning_image5.png

Tuning parameters of the material model

ALTRAN CZ is a leading provider of development services, especially for the automotive industry. With more than 25 years of experience with virtual design, part prototyping, and own accredited testing facility, it contributes to safe and effective individual transportation in Europe.

## Challenge
For the kinematics of the tripod manipulator, it is crucial to know how the constellation of actuators, based on their actual displacement, can affect the resulting position of the carried optical point in all six degrees of freedom ‐ all displacements and all rotations.

Moreover, the manipulator itself has to withstand all forces and stress resulting from the displacement of actuators. The tripod is also stabilized with three springs, each of them loaded with the prestress. Positions of these springs also may be changed during the design of the manipulator.

In total, there were 12 input parameters of the task, which is a massive extension to a previous study with only actuator displacements considered. For these three input parameters, a full-factorial scheme was used, leading to 27 computations in MSC.Nastran. However, this approach covered only the corners of the design space with a fixed interpolant level. Now, only 155 evaluations were needed for the expanded set of inputs and the creation of mathematical models for all ten outputs.

![tripod_manipulator_image4.png](https://uptimai-website.s3.eu-central-1.amazonaws.com/tripod_manipulator_image4_807e3bc814.png)

## Solution
In the beginning, we coupled the UptimAI tool with the simulation software MSC.Nastran, with no need to interfere with its code or features. Then, the UptimAI tool began to set combinations of inputs for the Nastran. It used the **UptimAI algorithm** to process simulation results into the metamodel of forces and displacements of the tripod manipulator.

Features of the mathematical model could be easily shown on the example of one of the outputs, the von Mises stress. **UptimAI sensitivity analysis** identified input parameters with the significant ability to change the output. Unlike most of the other outputs, the stress is dependent almost entirely on displacements of actuators. The effect of springs and their positions is too low to be measurable.

![tripod_manipulator_image1.png](https://uptimai-website.s3.eu-central-1.amazonaws.com/tripod_manipulator_image1_0f5936228f.png)

From the **UptimAI histogram**, the overall information about the range of this output was shown as well as the shape of the probability density function. Besides, it was possible to visualize the effect of each variable to the statistical properties of the model. Here the stress is ranging from 110 to 380MPa. From the influence analysis, it is apparent that setting the displacement of actuator 402 cannot lower the minimum value of stress below 110MPa. However, the probable maximum can be reduced by almost 30%.

![tripod_manipulator_image6.png](https://uptimai-website.s3.eu-central-1.amazonaws.com/tripod_manipulator_image6_50b5f0da70.png)

Details responses of the von Mises stress level to changes in input parameters were apparent from **UptimAI increment plots**. Depicted dependencies of the output on particular inputs are separate from their mutual interactions. The UptimAI algorithm was able to catch the discontinuity in behavior based on the actuator displacements. That increased significantly the precision of the mathematical model over the commonly used interpolation with the quadratic polynomials. The model itself then consists of the summation of its all available increments.

![tripod_manipulator_image3.png](https://uptimai-website.s3.eu-central-1.amazonaws.com/tripod_manipulator_image3_3524f1ab3e.png)

The statistical nature of the model and the performed analysis was also used for the definition of design recommendation. The **UptimAI preliminary optimization** tool was used to identify ranges of input parameters resulting in decreased stress values in the tripod. The result suggesting the lowest actuator displacement possible was not a big surprise. Nevertheless, it was the explicit mathematical confirmation of conclusions based on general engineering intuition.

![tripod_manipulator_image5.png](https://uptimai-website.s3.eu-central-1.amazonaws.com/tripod_manipulator_image5_fb87366ed9.png)

## Benefits
- **Deeper insight into the statistical effects** on tripod’s behavior.
- **Fast design exploration.** Set of only 155 FEM computations was sufficient to describe the problem with 12 input parameters and 10 outputs.
- **Simplified design task.** The number of input parameters can be reduced **while keeping the precision of the model**.

Influence analysis of the actuator displacements on stresses

small_tripod_manipulator_image2.png

thumbnail_tripod_manipulator_image2.png

tripod_manipulator_image2.png

Tripod manipulator

Commuter airplanes are typically used on short regional routes. They often operate in broken terrain of lands with underdeveloped infrastructure where they have to serve small regional airports with short landing strips.

## Challenge
Designers of STOL (Short Take-Off and Landing) airplanes had to reduce the landing ground distance to a minimum. To achieve the required aircraft performance, it was necessary to increase the maximum wing lift coefficient. Therefore, they had to design a new kinematic concept of a trailing-edge flap.

Due to the novelty of the concept, the performance of the new flap could not be assumed from measurements of well-known flap designs. Hence, it was necessary to understand how geometry and kinematics of the flap affect the maximum of the lift coefficient of the flapped airfoil. However, the standard methods of design exploration did not work, the development was delayed, and there was a failure to comply with deadlines.

There were 10 input parameters of the task. The standard approach of the Design Of Experiment required at least 1045 CFD computations in the Ansys FLUENT solver. In addition, achieving the quality of data interpolation and its sufficiency to mimic the real behavior of the flap aerodynamics was uncertain.

![stol_image6.png](https://uptimai-website.s3.eu-central-1.amazonaws.com/stol_image6_05e4a90fe8.png)

## Solution
In the beginning, we coupled the UptimAI tool with the simulation software Ansys FLUENT, with no need to interfere with its code or features. Then, the UptimAI tool begun to set combinations of inputs for the FLUENT. It used the **UptimAI algorithm** to process simulation results into the metamodel of the flapped airfoil aerodynamics.

Important general information about the performance of the chosen flap concept was given by UptimAI histograms. The probability distribution of the resulting increment in the maximal lift coefficient had the mean value 1.7, which refers to the very well-tuned slotted flap (closer to the typical Fowler flap performance). It also showed a fair probability of an increase in the lift coefficient of the flapped airfoil by more than 2.5! Therefore, it was confirmed that the proposed flap concept has the potential to improve the airplane’s flight characteristics.

![stol_image3.png](https://uptimai-website.s3.eu-central-1.amazonaws.com/stol_image3_375e2ef1bd.png)

**UptimAI histograms** also depicted particular probability distributions of results describing the influence of each input parameter and each interaction of input parameters. This helped to detect extreme values of the lift coefficient increments and their causes. A good example is the interaction of parameters describing the position of the pivot point, the flap deflection angle, and the deflection angle of the flap cove trailing edge. All three parameters together determine the size of the slot between the flap and the wing, which turned out to be crucial for the value of the lift coefficient.

![stol_image1.png](https://uptimai-website.s3.eu-central-1.amazonaws.com/stol_image1_00544fc558.png)

The importance of each input parameter in terms of aerodynamics was clearly shown in the **UptimAI sensitivity analysis**. It identified parameters emulating the additional camber of the flap (“4” – camber position, “9” – camber magnitude) as having just little influence on the lift coefficient. Concurrently they do not interact with other input parameters, thus, other criteria can be considered in their design, such as manufacturing. The same can be applied to the geometry of the flap cove entrance radius.

![stol_image2.png](https://uptimai-website.s3.eu-central-1.amazonaws.com/stol_image2_d8f77c965f.png)

Details of the dependence of the flap aerodynamics regard to changes in input parameters can be examined in **UptimAI increment plots**. Using these, designers were able to find ranges of inputs responsible for the steep drop in values of the lift coefficient which had to be avoided in the design to prevent deterioration of the result. This approach led to the design insensitive to manufacturing tolerances and deformations under loading. Increment plots were used as well for simulation points where the CFD solver failed to converge.

![stol_image4.png](https://uptimai-website.s3.eu-central-1.amazonaws.com/stol_image4_367221964d.png)

## Benefits
- **Increasing the maximum lift coefficient by up to 2.5** due to optimized flap geometry and position. The achieved properties significantly exceeded the defined expectations.
- **Landing ground distance shortened by up to 18%** compared to the original design with a slot flap.
- **Saving 450 hours of computational time.** The required number of CFD simulations (146) was only **14% of the original assumption.**


Landing ground distance shortened by up to 18%

small_stol_image5.png

thumbnail_stol_image5.png

stol_image5.png

Landing configuration of a STOL commuter airplane

The glide ratio, or the ratio between the lift and the drag coefficient, is one of the most important aerodynamic characteristics of an airfoil. It represents the aerodynamic efficiency of the airfoil and affects aircraft’s flight performance values such as the maximum range or angle of a climb after take-off.

## Challenge
The main goals for new airfoil design were to increase the glide ratio (efficiency) and decrease the sensitivity of airfoil’s aerodynamics to manufacturing tolerances and imperfections of its surface. Therefore, it was necessary to create a mathematical model describing dependencies between the glide ratio of the airfoil on its geometry.

The geometry was parameterized with a slightly modified PARSEC method. There were ten geometrical parameters in the task together with the Reynolds number describing flight conditions. Airfoil aerodynamics was analyzed in the panel method software Xfoil.

One computation in the Xfoil takes about 20 seconds; thus, this task was also suitable for validation of the mathematical model against the results of the Monte Carlo simulation. Result values interpolated from the metamodel were compared with the results of computations performed with the Xfoil.

![airfoil_design_image4.png](https://uptimai-website.s3.eu-central-1.amazonaws.com/airfoil_design_image4_68a60f11d4.png)

## Solution
In the beginning, we coupled the UptimAI tool with the panel method code Xfoil, with no need to interfere with its code or features. Then, the UptimAI tool began to set combinations of inputs for the Xfoil. It used the **UptimAI algorithm** to process computation results into the metamodel of the airfoil aerodynamics.

As shown in the **UptimAI histogram** and its probability density function, ranges of input parameters were set broad enough to conclude in the large variance of resulting values of the maximum glide ratio. It demonstrates possibilities of the chosen parameterization method and the capability of the Xfoil software to converge when solving aerodynamics of airfoils with multiple curvatures. A comparison of probability distribution of results, based on the glide ratio computed in Xfoil coincided well with the distribution of the modeled values of the maximum glide ratio, proving the accuracy of the metamodel. For the comparison, around 40,000 Xfoil computations were made in total with randomized values of input parameters (the Monte Carlo method). In contrast, the metamodel was created from only 161 Xfoil computations.

![airfoil_design_image3.png](https://uptimai-website.s3.eu-central-1.amazonaws.com/airfoil_design_image3_df7f58ff60.png)

**UptimAI sensitivity analysis** showed the influence of each input parameter on the maximum glide ratio. It confirmed the major effect of inputs describing the upper surface of the airfoil, especially its maximum thickness. On the other hand, the influence of the trailing edge taper was lower than expected, but this could be a result of the defined range of this input parameter. The leading edge radius of the upper surface did not affect the glide ratio either.

![airfoil_design_image2.png](https://uptimai-website.s3.eu-central-1.amazonaws.com/airfoil_design_image2_0177ea072b.png)

Then, direct response of the maximum value of the glide ratio to changes in each of the influential inputs was examined using the **UptimAI increment plot**. The shape of these functions was key to the airfoil design. This can be demonstrated on the position of the maximum thickness on the airfoil’s upper surface. For this input parameter, the range of values could be identified with a low response in result values as well as the region with a steep gradient of the output. For the first range, the glide ratio becomes insensitive to surface imperfections. The second one leads to significant aerodynamic improvements, only when the manufacturing of the airfoil is flawless.

![airfoil_design_image1.png](https://uptimai-website.s3.eu-central-1.amazonaws.com/airfoil_design_image1_0289314682.png)

## Benefits
- **Deeper insight into the statistical effect** on the airfoil aerodynamics.
- **Increase in the average value of the glide ratio from 113 to 166**. All of the newly designed airfoils have a maximum glide ratio higher than 125.
- **Validation of the metamodel** – probability distribution of interpolates results coincide with results computed in Xfoil.


small_airfoil_design_image5.png

thumbnail_airfoil_design_image5.png

airfoil_design_image5.png

Airfoil Design

ŠKODA AUTO is the largest car manufacturer in the Czech Republic and part of the Volkswagen Group since 1991. The company uses highly advanced simulation tools in its R&D departments to meet all design goals, where passenger safety is one of the top priorities.

## Challenge
During the development of a new car model, it was necessary to increase the amount of energy absorbed during a crash incident. Thus, engineers had to learn and understand how can the geometry of certain body parts affect the amount of energy needed for deformation of these parts. To validate the new know-how before its use for car design, results of performed simulations had to be compared against the real testing. However, test results were prone to correct settings and adjustment of the testbed. The task had 27 input parameters in total. Standard approaches of design exploration would require at least 4000 runs of the PAM-Crash simulation software, making the study unfeasible. On the other hand, when reduced, the study would fail to give complete information about the behavior of the car part and relations between inputs and the absorbed energy.

![boxbeam-car.png](https://uptimai-website.s3.eu-central-1.amazonaws.com/boxbeam_car_b7eceeb94e.png)

## Solution
In the beginning, we coupled the UptimAI tool with the simulation software PAM-Crash, with no need to interfere with its code or features. Then, the UptimAI tool begun to set combinations of inputs for the PAM-Crash. It used the UptimAI algorithm to process simulation results into the metamodel of the tested part under deformation.

There were geometrical input parameters in the task meant to be adjusted to increase the performance of the part. Once the metamodel was ready, UptimAI identified ranges of selected parameters raising the probability of an increase in the energy absorbed by the deformed part. At the same time, these new ranges were designed to decrease the variance of results as much as possible. This approach diminishes the influence of uncertainties in the rest of the inputs.

![boxbeam_1_EN.PNG](https://uptimai-website.s3.eu-central-1.amazonaws.com/boxbeam_1_EN_4cebbd013b.PNG)

**UptimAI sensitivity analysis** and **UptimAI histograms** helped to identify input parameters with a major influence on both, the variance and the mean value of the amount of energy absorbed during part deformation. It was revealed that the variance of results is strongly affected by barrier rotation – the inclination angle between the testbed impactor and the tested part. Thus, one of the optimization goals was the minimization of the influence of this particular parameter.

![boxbeam_2_EN.PNG](https://uptimai-website.s3.eu-central-1.amazonaws.com/boxbeam_2_EN_a2b90ba17f.PNG)

Another guidance to understanding the problem of the deformed box beam was given by the **UptimAI increment plot**. It showed shapes of increment functions depicting dependencies and relations of input parameters and the value of absorbed energy. To reduce the influence of uncertainties on the testbed, UptimAI recommended setting ranges of inputs with the lowest gradient of corresponding increment functions. One example – material quality check of used sheet metals is one of the ways how to diminish the influence of the impactor rotation.

![boxbeam_3_EN.PNG](https://uptimai-website.s3.eu-central-1.amazonaws.com/boxbeam_3_EN_f4acf54d6e.PNG)

The UptimAI software also found input parameters connected with others through higher-order interactions. Then, engineers used the **UptimAI preliminary optimization** tool to identify ranges of inputs resulting in increased absorbed energy and also ranges to be avoided. A good example of such output is the recommendation to choose distinct thicknesses of two sheet metals welded together to form the tested part. There was a larger amount of absorbed energy in comparison to the case with both sheet metals being thicker. Additionally, the results of an older study recommending non-equidistant positions of spot welds were confirmed.

![boxbeam_4_EN.PNG](https://uptimai-website.s3.eu-central-1.amazonaws.com/boxbeam_4_EN_cb670c85b8.PNG)

## Benefits
- **The absorbed impact energy of optimized geometries increased by 10% on average**, the lowest value (absorbed energy due to the most adverse combination of inputs) increased by 18%.
- Decrease of result sensitivity to testing conditions – **the variance of the results decreased by 63%**.
- **Significant reduction of the computational time** – necessary number of simulation runs (280) was reduced to **7% of the original estimate**.


Case studies