Signpost the Future: Simultaneous Robust and Design Optimization of a Knee Bolster

Transcription

1 Signpost the Future: Simultaneous Robust and Design Optimization of a Knee Bolster Tayeb Zeguer Jaguar Land Rover W/1/012, Engineering Centre, Abbey Road, Coventry, Warwickshire, CV3 4LF tzeguer@jaguar.com Stuart Bates Altair ProductDesign Imperial House, Holly Walk, Royal Leamington Spa, CV32 4JG Andy.burke@uk.altair.com copyright Altair Engineering, Inc. 2011

2 Abstract The future of engineering design optimization is robust design optimization whereby a design is optimized for real world conditions and not just for one particular set of ideal conditions (i.e. nominal). There is no practical point trying to get to the peak of a mountain to get the best view when a slight gust of wind can blow you off, what is practical is to find the highest plateau where the view is unaffected. The same is true for engineering design, there is no point in coming up with a design which is optimized for a set of ideal conditions when in reality there exists uncertainty in the materials, manufacturing and operating conditions. This paper introduces a practical process to simultaneously optimize the robustness of a design and its performance i.e. finds the plateau rather than the peak. The process is applied to two examples, firstly to a composite cantilever beam and then to the design of an automotive knee bolster system whereby the design is optimized to account for different sized occupants, impact locations, material variation and manufacturing variation. Keywords: Optimization, HyperStudy, Stochastic, Uncertainty, LS-DYNA 1.0 Introduction The competitive nature of the automotive industry demands continual innovation to enable significant reductions in the design cycle time while satisfying ever increasing design functionality requirements (e.g. minimising mass, maximising stiffness etc). The challenges for computer-aided engineering (CAE) to overcome are: Development cycle must be reduced. Failure modes have to be found and resolved earlier. The enablers for CAE are: Faster model creation, CPU, Automation, Material property Identification, Robustness Optimisation and Validation. The aim of this work is to show that Altair HyperStudy [1] can be used as powerful CAE enabler to facilitate robust design. Over the last decade industry has been indoctrinated into the philosophy of manufacturing quality to six sigma. This paper presents increasing applications of designing systems to sigma levels of quality. Thus ensuring that designs or numerical models perform within specified limits of statistical variation. Copyright Altair Engineering, Inc.,

3 Robust Optimized Design Figure 1: Design for Six-Sigma Process An overview of each stage of the Design for Six Sigma (DFSS) process is given below. 1.1 Define The first step is to carry out brainstorming to define the system inputs, outputs, controllable and uncontrollable factors. The Parameter Diagram or p-diagram (Figure 2) is a useful tool for such a purpose. Uncontrollable Factors INPUT Performance Targets DESIGN OUTPUT Performance Controllable Factors 1.2 Characterize Figure 2: Define P-Diagram The characterization phase involves the following :- Copyright Altair Engineering, Inc.,

4 Key parameter identification: identifies the parameters which have the most significant effect on the performance (output) of the design. This is done typically through the use of design of experiments (DoE) and statistics (e.g. analysis of variance ANOVA). Surrogate Model generation: Typically, in CAE the analysis of a non-linear design will require simulation times ranging from one hour to a day, making the use of full analyses for iterative design optimisation computationally expensive and a robustness assessment requiring hundreds or thousands of Monte Carlo simulations impractical. To overcome these problems a response surface approximation or surrogate model is required. This is done using the information generated by the DoE together with advanced surface-fitting algorithms. The surrogate model gives the value of a key output variable in the design space, e.g. peak deceleration, as a function of the design variables. Thousands of simulations of the surrogate model can be run in a few minutes. 1.3 Optimize Figure 3 shows a typical design space (response surface) for two design variables. If you assume there is no variation in the operating and manufacturing conditions then point A is the optimum solution. However, in reality there are variations in the manufacturing and operating conditions such that it is very easy to fall off this optimum point (A). A better or robust optimum is point B since the design space is flatter in that region i.e. the performance of the design is less sensitive to real life variations. The aim of this optimization phase is to identify the most robust solution in the design space. SIMPLE OPTIMUM POINT Absolute highest peak ignored due to sharp gradients surrounding it, reflecting the non-robust nature of the solution A small change in input (X or Y) will result in a rapid change in output (Z) ROBUST OPTIMUM CLOUD Peak B has value lower than Peak A The flatter landscape in the region of the peak results in more robust solutions in that area The output (Z) will not be highly sensitive to small changes X or Y Figure 3: Robust Optimum Identification The process developed here is shown in Figure 4 and consists of the following three stages: Copyright Altair Engineering, Inc.,

5 Stage 1: Assessment & Optimization of the baseline design performance under ideal conditions (i.e. deterministic optimization). This enables a rapid judgment as to whether an improved/feasible design exists within the bounds of the design i.e. for the initial structural layout within the allowable thickness ranges. Stage 2: Robustness assessment: assessment of the mean and variation in the performance of a design when subjected to real conditions. Stage 3: Optimization under real conditions (robustness optimization) simultaneously optimize the mean and variation of performance when subjected to real life variations. Previous studies have performed deterministic optimization followed by robustness assessments [2]. However, this study presents the first HyperStudy applications of simultaneous robust optimization. Baseline Design Stage 1 Design Assessment & Optimization Under Ideal Conditions Suitable Design? No Yes Stage 2 Design Assessment Under Real Conditions Robust Optimized Design Stage 3 Robustness Optimization: Design Optimization Under Real Conditions 1.4 Verify Figure 4: Simultaneous Robust and Design Optimization Process The staged optimization process (section 1.3) provides invaluable sensitivity data in order to understand which variables are driving the robustness or optimization of the system. This inevitably will produce better design. In addition, since a consistent virtual environment is used for all three stages of this optimization process, a high degree of self checking is automatically performed. However, the true verification of the process is the production of the physical design which exhibits a robust performance in any experimental testing programme and ultimately reduced warranty claims from the field. The methodology for generating optimal robust designs that has been developed in this work is primarily focused on the optimize phase of the DFSS loop (Figure 1). It is described through use of two examples described in Sections 2 & 3. The first is a composite cantilever Copyright Altair Engineering, Inc.,

6 beam, on which the methodology was developed and the second is an industrial example: the design of a knee bolster system. 2.0 Composite Beam Design This example is concerned with the minimization of the weight of a cantilevered composite beam (Figure 5) subjected to a parabolic distributed load (q) with uncontrollable and controllable factors such as manufacturing or material variation. The DFSS process has been applied to the problem and is described below. Figure 5: Composite Beam Subjected to a Parabolic Distributed Load 2.1 Define Figure 6 shows the p-diagram for the composite beam. The performance targets for the beam are as follows: Deflection at the free end of the beam < 1 (normalized). Maximum bending stress in the beam < 1 (normalized). Height < 10 times the width (to avoid torsional lateral buckling). NOISE fiber volume fraction ± 0.03 Young s modulus of the fiber ± 2% Young s modulus of the resin ± 2% Density of the fiber ± 2% Density of the resin ± 2% Width ± 0.3mm Height ± 0.3mm INPUT Deflection & Stress Targets COMPOSITE BEAM OUTPUT Max Deflection, Max Bending Stress, Height to width ratio PARAMETERS Beam Height Beam Width Fibre Volume Fraction Figure 6: P-Diagram for the Composite Beam Copyright Altair Engineering, Inc.,

7 2.2 Characterize The key parameters for the beam and their variations are as given in the p-diagram in Figure 6. The analysis of the beam is via an analytical expression, as such there is not a requirement to replace the analysis with a surrogate model as is the case for the knee bolster analysis in Section Optimize Stage 1: Design Assessment & Optimization Under Ideal Conditions Typically, during an engineering design process once a baseline design has been generated (e.g. from a topology optimization) it is assessed to determine whether or not it meets the performance criteria. The baseline design performance is given in Table 2, it can be seen that the design meets the targets and has a weight of 4.8N. The next stage is to determine the minimum weight design which meets the targets. In order to reduce complexity, ideal conditions are assumed at this stage and optimization is carried out on perturbations of the initial structural layout and thicknesses. This stage rapidly provides information as to whether or not an improved/feasible design exists within these design bounds. The engineer can then make a judgment as to whether or not the design is suitable for further development and can be taken forward to stage 3 or if a modified baseline design is required. The optimization of the beam is set up is as follows: Objective: o Minimize Weight Constraints: o Maximum deflection at the free end of the beam (normalized) < 1 o Maximum bending stress in the beam (normalized) < 1 o Height to 10 x Width ratio (normalized) < 1 (to avoid torsional lateral buckling) Design Variables: o 4mm Beam Width 20mm o 20mm Beam Height 50mm o 0.4 Fibre Volume Fraction 0.91 Altair HyperStudy is used for the optimization and the results are shown in Table 1. It can be seen that, the optimum design (for ideal conditions) meets the targets and represents a 39% weight reduction over the baseline design. Copyright Altair Engineering, Inc.,

8 Design variables Min Max Baseline design Optimum Under Ideal Conditions width [mm] height [mm] Fibre volume fraction Objective (min): weight [N] Constraints Normalized stress constraint <= Normalized displacement constraint <= Normalized Height to 10 x width ratio <= Table 1: Performance of the Baseline and Optimum Designs Under Ideal Conditions Stage 2: Design Assessment - Under Real Conditions At this stage, the design is subjected to variations in the uncontrollable/controllable factors present in a real system. The mean and variation of the performance is assessed via a stochastic study in HyperStudy. For the beam example the variations imposed on the design are material and manufacturing tolerances. Note, the variations are assumed to be normally distributed and ±3σ covers the interval of the tolerance where σ is the standard deviation of the distribution. Table 2 identifies the tolerances and their assumed variations. Material related tolerances Variation fiber volume fraction ± 0.03 Young s modulus of the fiber ± 2% Young s modulus of the resin ± 2% Density of the fiber ± 2% Density of the resin ± 2% Geometric related tolerances Width Height ± 0.3mm ± 0.3mm Table 2: Variations on Manufacturing and Material Tolerances The mean and variation (σ) in the performance of a design is determined by executing a 10,000 Monte Carlo (MC) simulation run using a random Latin Hypercube DoE (RLH) (Figure 7(a)) on the design with the imposed variations listed in Table 1. Note, a 500 MC simulation run (Figure 7(b)) was also carried out and the resulting statistics were similar to the 10,000 MC simulation as can be seen in Table 3, therefore for a more computationally expensive Copyright Altair Engineering, Inc.,

9 analysis it can be reasonably assumed that the resulting statistics (using a RLH) will be practically the same with a reduced number of runs. (a) 10,000 runs (b) 500 runs Figure 7: Comparison of Monte Carlo Simulation Run Plots for the Baseline Design Stochastic Assessment Mean Standard Deviation 500 runs runs 500 runs runs weight [N] Normalized stress Normalized displacement Normalized height to width ratio Table 3: Comparison of Statistics for the Monte Carlo Simulations on the Baseline Design The results of the stochastic studies carried out on the baseline and deterministic optimum designs are given in Figure 8 and Table 4. Each point on the plots represents a run in the MC simulation and the resulting cloud of points gives the resulting mean and variation in performance of a particular design. The green circle (Figure 8) represents the boundary of 3σ i.e. 3-sigma design. Hence, if an engineer is aiming for a 3-sigma performance (99.73 % reliability) then this circle must lie in the feasible region. It can be seen, that the clouds for both the baseline and deterministic optimum designs are centred on the point where the stress and displacement = 1 i.e. the mean performance is the target value of 1, however it can also be seen that approximately 75% of the runs for both designs are infeasible since their values >1 i.e. the 3σ boundary lies in the infeasible zone. Note also, that the cloud for the deterministic optimum has a greater scatter than the baseline i.e. it is less robust since it s variation in performance is greater. As a result neither design can be considered as robust. Copyright Altair Engineering, Inc.,

10 2.3.3 Stage 3: Simultaneous Robustness Optimization Under Real Conditions In order for a design to be simultaneously robust and optimized the centre of the performance cloud (i.e. mean performance) must be as close to the constraint boundaries as possible whilst ensuring that, for 3-sigma performance, the 3-sigma boundary remains in the feasible region i.e % of the points in the cloud are in the feasible region. Similarly, for 6-sigma designs the 6-sigma boundary remains in the feasible region. The robustness optimization of the beam for 3-sigma performance is set up is as follows (note the mean and σ are calculated as in Stage 2) and carried out using HyperStudy. Objective: o Minimize Mean Weight Constraints: o σ weight 3σ (assume σ weight = 0.1) o Mean Normalized Stress + 3σ 1 (assume σ stress = 0.1) o Mean Normalized Displacement + 3σ 1 (assume σ disp = 0.1) o Mean Normalized height to width ratio + 3σ 1 (assume σ h2w = 0.1) Design Variables: o 4mm Beam Width 20mm o 20mm Beam Height 50mm o 0.4 Fibre Volume Fraction 0.91 where σ is the standard deviation. The results of the robustness optimization are given in Figure 8 and Table 4. The robust optimum represents a 29% weight reduction over the baseline design. It can be seen, that the cloud for the robust optimum design is centred within the feasible stress-displacement region and the 3σ boundary lies in the feasible zone. Baseline Deterministic Optimum Robust Optimum Figure 8: Results of the Stochastic Studies Indicates 3 sigma boundary Copyright Altair Engineering, Inc.,

11 Baseline Deterministic Robust Design variables Min Max design Optimum Optimum Mean width [mm] Mean height [mm] Mean Fiber volume fraction v f Objective (min): Mean Weight [N] Constraints Mean Normalized stress constraint + 3 sigma <= Mean Normalized displacement constraint + 3 sigma <= Mean Normalized Height to 10 x width ratio + 3 sigma <= meets targets fails targets Table 4: Performance of Baseline, Deterministic Optimum and Robust Optimum 2.4 Verify Since all of the performance calculations are carried out using the full analysis of the beam i.e. an analytical equation, the verification phase is completed at the optimization stage. 3.0 Knee Bolster Study 3.1 Introduction The aim of this study was to apply the same process as in Section 2 to determine a robust and optimized design of a knee bolster. The study has been carried out on a sub-system model of the knee bolster (Figure 9a). The dynamic finite element analysis code LS-DYNA [3] was used to compute the response of the system to various design inputs. The objective of the study was to automatically vary various design variables to optimize the energy absorbing characteristics of the system whilst satisfying various force and displacement limiting constraints based on federal requirements: FMVSS 208 [4]; final verification was carried out using full occupant / interior model simulations using LS-DYNA (Figure 9b). Copyright Altair Engineering, Inc.,

12 Knee Bolster (a) Sub-System Model (b) Full Model Figure 9: LS-DYNA Analysis of the Knee Bolster Design The Design for Six-Sigma (DFSS) process (Section 1) has been applied to the knee bolster design as is described in this section. 3.2 Define The knee bolster system is defined through the p-diagram shown in Figure 10. NOISE Material Yield Stress Manufactured Thickness Manufactured Shape Impactor type (5 th %, 50 th %) Impactor position variation INPUT FMVSS208 USNCAP EURONCAP Knee Bolster OUTPUT Force-displacement Pulse PARAMETERS Thickness Shape Material Properties Impactor position P-Diagram Figure 10: P-Diagram for the Knee Bolster System It can be seen, that the inputs are the legislative targets for the system which are based on the force-displacement and energy absorption of the knee bolster. Hence the output is the forcedisplacement pulse measured from the LS-DYNA simulation. The targets for the knee bolster Copyright Altair Engineering, Inc.,

13 are that the normalized force and displacement values are less than 1. The normalization is done according to FMVSS 208 [4]. A set of typical force-displacement pulses for the 5 th Left/Right & 50 th Left/Right impactors is shown in Figure 11. It can be seen, that this solution is feasible since the corresponding normalized force and displacement values are less than one i.e. in the feasible region. Feasible Region Figure 11: Typical Force-Displacement Output The thickness and shape parameters are identified in Figure 12. The thickness ranges are assumed to vary between 1 and 10mm. The shape factor varies between -1 and 1. Figure 13 shows the assumed variation of ±25mm in the centre point of the 5 th and 50 th impactors. Copyright Altair Engineering, Inc.,

14 PARAMETERS Thickness Shape Material Properties Impactor position Thickness 1 Thickness 2 Thickness 3 Thickness 4 Shape Variable Note: the thickness and shape variables are the same for each knee bolster Figure 12: Thickness and Shape Parameters for the Knee Bolster PARAMETERS Thickness Shape Material Properties Impactor position (a) 5 th Percentile Impactors (b) 50 th Percentile Impactors Figure 13: Impactor Position Variation Copyright Altair Engineering, Inc.,

15 Contributing % Impactor position vertical Impactor position Horizontal Shape Thickness 1 Thickness 2 Thickness 3 Thickness 4 Yield Stress Characterize The next stage was to identify the key parameters which have the greatest effect on the knee bolster performance. This was done using Altair HyperStudy using the following process: 1 Run a DoE with all the parameters 2 Create an approximation of the responses 3 Carry out a statistical analysis of the approximation using Analysis of Variance (ANOVA) Figure 14 shows the results of the ANOVA study for the displacement of the 5 th left Impactor. This is typical of the results for the other responses. It can be seen, that the position of the impactors, the shape and thickness variables and the yield stress contribute the most to the response. It is assumed that changes to these parameters are sufficient to characterize the knee bolster system. 20 ANOVA plot % Contributions of the Parameters to Displacement of 5 th Left Impactor Other less significant parameters 0 Contributing Source Figure 14: Key Parameter Identification Typical ANOVA Plot Following on from this, a response surface of the LS-DYNA analysis was generated for use in the optimization phase. There are a number of possibilities available in HyperStudy for doing this. However, the recommended approach (used here) is to carry out a DoE study using the optimal design filling algorithm Optimal Latin Hypercube, and then use this data to create a surrogate model via the moving least squares method. Figure 15 shows a typical response surface generated for the force response in the 50 th left impactor. Copyright Altair Engineering, Inc.,

16 Force 50 th Left Typical Response Surface Impactor Position Horizontal Impactor Position Vertical Figure 15: Typical Response Surface With the knee bolster system define and characterized the next step is then to optimize the design. 3.4 Optimize As described earlier the optimize phase has 3 stages which are shown in Figure 4, these are described in this section Stage 1: Design Assessment & Optimization Under Ideal Conditions The first stage is to assess and optimize the design under ideal conditions i.e. no noise is imposed on the system. Therefore, the only parameters under consideration are the thickness and shape variables (Figure 12). The response surface generated in the characterization phase is used for the analysis. The setup is as follows: Objective: o Maximize Sum Normalized Energies Constraints: o Normalized Force: 1.0 o Normalized Displacement: 1.0 Design Variables (Figure 12): o 1mm 4 Thicknesses 10mm o -1 Shape variable Scale Factor 1 Copyright Altair Engineering, Inc.,

17 The optimization is carried out using the gradient-based optimizer in Altair HyperStudy. The results are given in Table 5 and Figure 16, it can be seen that for ideal conditions the solution meets the performance targets. The question arises at this point: how does this solution behave in reality? This is addressed in the next section. Design variables Min. Max Design Optimized for IDEAL conditions Shape Variable Thickness 1 [mm] Thickness 2 [mm] Thickness 3 [mm] Thickness 4 [mm] Objective (max): Sum of Normalized Energy Constraints 5th 50th left right left right Normalized displacement constraint <= Normalized force <= Table 5: Assessment of the Design Optimized for Ideal Conditions 5 th Left 5 th Right 50 th Left 50 th Right Figure 16: Assessment of the Design Optimized for Ideal Conditions Copyright Altair Engineering, Inc.,

18 3.4.2 Stage 2: Design Assessment - Under Real Conditions In order to assess real life performance a robustness assessment (stochastic study) of the design optimized for ideal conditions is carried out. This is done with a Monte Carlo simulation carried out on the response surface, here a 500 run random Latin Hypercube is used. The parameters and assumed real life variations imposed on the system are identified in Table 6. Note, the following assumptions have been made: the variations are normally distributed and ±3σ covers the interval of the tolerance where σ is the standard deviation of the distribution. Material related tolerances Variation Yield Stress ± 10% Geometric related tolerances T hickness ± 0.1mm Shape Scale Factor ± 0.01 Impactor position variation Position ± 25mm Table 6: Knee Bolster Noise Parameters and Variations The results of the robustness assessment performed on the design optimized under ideal conditions are shown in Figure 17. It can be seen from the resulting performance clouds that there are a large number of solutions which fail the force performance targets and the design is considered non-robust. Copyright Altair Engineering, Inc.,

19 5 th Left 5 th Right 50 th Left 50 th Right Figure 17: Design Optimized for Ideal Conditions - Robustness Assessment Stage 3: Robustness Optimization: Design Optimization Under Real Conditions At this stage the robustness assessment is incorporated in the optimization loop. The output from the robustness assessment used in the optimization loop is the mean and standard deviation of the responses. The optimization is set up as follows: Objective: o Maximize Mean of the Summed Normalized Energies Constraints: o Normalized Displacement: Mean + 3σ 1.0 o Normalized Force: Mean + 3σ 1.0 Design Variables (Figure 12): o 1mm 4 Thicknesses 10mm o -1 Shape variable Scale Factor 1 The results of the simultaneous robustness and design optimization are shown in Figure 18. It can be seen the performance clouds have been shifted into the feasible region, Although there are a small number of solutions which fail the performance targets, the design is considered as robust as possible for the current knee bolster structural layout. Copyright Altair Engineering, Inc.,

20 5 th Left 5 th Right 50 th Left 50 th Right Figure 18: Design Optimized for Real Conditions - Robustness Assessment 3.5 Verify At this stage of the DFSS process significant information about the performance of the knee bolster has been generated. The next step is then to plug the design back into the full vehicle model which has been concurrently updated with other optimized components of the car. It is a design challenge to produce a virtual design that can achieve the constraint targets within ±3σ due to the conservative nature of this numerical test environment (e.g. totally rigid backing structure, conservative impact velocity etc.). This technology can be efficiently used to determine the most efficient design for the specified design variations. The design determined by this process is similar to a production component used on a recent vehicle. However, this design was achieved in a fraction of the design time with an increased understanding of the performance drivers. Copyright Altair Engineering, Inc.,

21 4.0 Conclusions The future of engineering design optimization is robust design optimization whereby a design is optimized for real world conditions and not just for one particular set of ideal conditions. There is no point in coming up with a design which is optimized for a set of ideal conditions when in reality there exists uncertainty in the materials, manufacturing and operating conditions. Altair HyperStudy has been used to simultaneously optimize the robustness and performance of a real world component (i.e. automotive knee bolster). The resulting design was similar to an existing production component. However, this design was achieved in a fraction of the design time with an increased understanding of the performance drivers. A unique process has been developed which is computationally efficient for complex non-linear systems. This process can be further enhanced and automated. The study has shown that Altair HyperStudy can be used as a key CAE enabler. Achieving robust design is inherent in the quality philosophy of many companies. It will become an increasing requirement to demonstrate that digital designs achieve the required quality levels. This will initially be achieved on a component level and gradually migrate to complex systems. The initial requirement will be to understand the probabilistic variation of various parameters. This will require an increasing amount of measurement and an increased understanding of the physical drives of the component / system. Robustness can only be achieved by understanding the variation of the various factors. Adding noise factors during optimisation is the best way in obtaining a robust solution the use of DFSS principle helps identify failure modes and eliminate them earlier in the design process. For certain parameters, suppliers are already instructed to deliver product within specific sigma quality levels. This technology can identify parameters which drive the quality and help develop guidelines to control the variation of these quantities. This control will be accompanied by an associated cost penalty. Increased availability of inexpensive powerful computing and improvements to software integration and the predictive algorithms heralds the new development of producing digital designs to sigma levels of quality. 5.0 References [1] Altair HyperStudy 8.0 Altair Engineering Inc. (2006). [2] Design Optimization and Probabilistic Assessment of a Vented Airbag Landing System for the ExoMars Space Mission, Richard Slade and Andrew Kiley, 5 th Altair UK Technology Conf., April Copyright Altair Engineering, Inc.,

22 [3] 'LS-DYNA Version 970, Livermore Software Technologies Corporation, LSTC Technical Support, [4] FMVSS 208 Occupant Crash Protection, Federal Motor Vehicle Safety Standards and Regulations. Copyright Altair Engineering, Inc.,