Successive Response Surface Method Research Articles

A Multi-Fidelity Successive Response Surface Method for Crashworthiness Optimization Problems

Due to the high computational burden and the high non-linearity of the responses, crashworthiness optimizations are notoriously hard-to-solve challenges. Among various approaches, methods like the Successive Response Surface Method (SRSM) have stood out for their efficiency in enhancing baseline designs within a few iterations. However, these methods have limitations that restrict their application. Their minimum iterative resampling required is often computationally prohibitive. Furthermore, surrogate models are conventionally constructed using Polynomial Response Surface (PRS), a method that is poorly versatile, prone to overfitting, and incapable of quantifying uncertainty. Furthermore, the lack of continuity between successive response surfaces results in suboptimal predictions. This paper introduces the Multi-Fidelity Successive Response Surface (MF-SRS), a Gaussian process-based method, which leverages a non-linear multi-fidelity approach for more accurate and efficient predictions compared to SRSM. After initial testing on synthetic problems, this method is applied to a real-world crashworthiness task: optimizing a bumper cross member and crash box system. The results, benchmarked against SRSM and the Gaussian Process Successive Response Surface (GP-SRS)—a single-fidelity Gaussian process-driven extension of SRSM—show that MF-SRS offers distinct advantages. Specifically, it improves upon the specific energy absorbed optimum value achieved by SRSM by 14%, revealing its potential for future applications.

Properties of Rubber-Like Materials and their Blends in Wide Range of Temperatures – Experimental and Numerical Study

Abstract Elastomers are widely used in many industries. Their use requires thorough knowledge of their strength and stiffness parameters over a wide temperature range. However, determination of the parameters of such materials is still a challenge. Therefore, the paper presents research methodology allowing determination of the properties of rubber-like materials in a wide range of stretch and temperatures (from +50°C to −25°C) by using the example of styrene-butadiene rubber (SBR) and natural rubber (NR) elastomers. Additionally, two blends, chloroprene rubber/nitrile-butadiene rubber (CR/NBR) and NR/SBR blends, were also considered. Based on physical premises, a polynomial and Arruda–Boyce hyperelastic constitutive models parameters were determined using two different methods, namely curve-fitting and the successive response surface method.

The Linear Interpolation Approach (LInA), an approach to speed up the Successive Response Surface Method

Metamodel-based approaches are very often used for nonlinear optimization. The Successive Response Surface Method (SRSM) (Roux et al. in Int J Numer Methods Eng 42:517–534, 1998; Stander and Craig in Eng Comput 19:431–450, 2002) is one example for the application of such an approach. SRSM computes a smoothing approximation for each system response in each iteration based on a set of sample points generated in a defined domain around the current optimum. This domain is shifted and contracted in each iteration according to the resulting current best point. By default, a linear polynomial is used as the approximation function. A new Linear Interpolation Approach (LInA) has been developed to reduce the number of nonlinear analyses required during the optimization. In contrast to SRSM, LInA uses interpolating linear polynomials rather than regressions. Linear approximation is used to exclude artificial local minima between sample points. Due to the fact that no oversampling is required anymore, interpolation results in a lower number of nonlinear analyses without losing accuracy. A second reduction of the required analyses is achieved by avoiding a complete resampling in each iteration. It is assumed that the vectors defined from the current best point to the remaining sample points should be as orthogonal as possible. Therefore, LInA starts with initial sample points from a Koshal design of experiments (DOE) locally around the initial design point. After having computed the new best point based on the linear approximation, only a few existing sample points are deleted in each iteration and replaced by new ones. Complete QR decomposition is used to generate the new sample points in an orthogonal complementary subspace. First studies with two nonlinear structural optimization problems have shown that the LInA approach leads to a noteworthy reduction of the number of required nonlinear analyses in comparison to SRSM. Furthermore, LInA is compared to two kriging approaches using a subdomain similar as LInA does. LInA shows results comparable to kriging with respect to the number of analysis runs.

Six sigma robust design optimization for thermal protection system of hypersonic vehicles based on successive response surface method

Lightweight design is important for the Thermal Protection System (TPS) of hypersonic vehicles in that it protects the inner structure from severe heating environment. However, due to the existence of uncertainties in material properties and geometry, it is imperative to incorporate uncertainty analysis into the design optimization to obtain reliable results. In this paper, a six sigma robust design optimization based on Successive Response Surface Method (SRSM) is established for the TPS to improve the reliability and robustness with considering the uncertainties. The uncertain parameters related to material properties and thicknesses of insulation layers are considered and characterized by random variables following normal distributions. By employing SRSM, the values of objective function and constraints are approximated by the response surfaces to reduce computational cost. The optimization is an iterative process with response surfaces updating to find the true optimal solution. The optimization of the nose cone of hypersonic vehicle cabin is provided as an example to illustrate the feasibility and effectiveness of the proposed method.

Automatic design of marine structures by using successive response surface method

An automated optimization procedure based on successive response surface method is presented. The method is applied to weight optimization of a stiffened plate used in marine structures. In the des...

Finite Element Modelling of a Hybrid III Dummy and Material Identification for Validation

Anthropomorphic test devices (ATDs) are used to predict the human injury risk in a crash test. The Hybrid III crash test dummy is a standard ATD used for measuring the occupant safety in a frontal impact test. Since a real crash test using a vehicle is expensive, computer simulation using the finite element method (FEM) is widely used. Therefore, a detailed and robust finite element (FE) dummy model is required to acquire more precise occupant injury data and more accurate behaviour of a dummy during the crash. This research proposes the process of modelling and validation of an FE model of a Hybrid III crash test dummy, and a high-fidelity FE model of the Hybrid III dummy is developed on the basis of the proposed process. The proposed process consists of two phases. First, the FE model is constructed on the basis of the reverse engineering technique. Second, the material identification process is performed for validation of the constructed FE model of the Hybrid III crash test dummy. A certification test for each part of the physical dummy model and also computer simulation of the constructed FE model are performed. The material identification process is carried out by an optimization process to minimize the difference between the physical test results and the computer simulation results. The utilized optimization algorithm is the successive response surface method (SRSM).

Bus rollover crashworthiness under European standard: an optimal analysis of superstructure strength using successive response surface method

Bus rollover is one of the most serious of accidents. Strengthening bus frames to maintain survivor space and reduce occupant injury is necessary following the issue of Regulation No. 66 by the Economic Commission for Europe (ECE R66). Whilst increasing bus weight is unlikely because of production costs and fuel economy, this paper presents an optimal method of bus rollover crashworthiness design. In this study a full-scale, validated, finite element (FE) model of the vehicle was used. Optimisation was performed by the successive response surface method (SRSM) with LS-OPT, a design variable analysis method based on the parameterisation of the energy absorption ability of bus frame components. LS-DYNA was used as the FE solver. An optimal prototype of the vehicle was obtained with crashworthiness following ECE R66 and vehicle weight at the existing level was maintained with the improvement in the lower displacement by 49.2% and upper displacement by 39.4% of bus frames versus bus survivor space. This paper presents a procedure for bus rollover crashworthiness design related to vehicle weight, with a robust and effective method using an optimal technique combining LS-DYNA and LS-OPT.

A Computationally Efficient Formal Optimization of Regional Myocardial Contractility in a Sheep With Left Ventricular Aneurysm

A non-invasive method for estimating regional myocardial contractility in vivo would be of great value in the design and evaluation of new surgical and medical strategies to treat and/or prevent infarction-induced heart failure. As a first step towards developing such a method, an explicit finite element (FE) model-based formal optimization of regional myocardial contractility in a sheep with left ventricular (LV) aneurysm was performed using tagged magnetic resonance (MR) images and cardiac catheterization pressures. From the tagged MR images, 3-dimensional (3D) myocardial strains, LV volumes and geometry for the animal-specific 3D FE model of the LV were calculated, while the LV pressures provided physiological loading conditions. Active material parameters (T(max_B) and T(max_R)) in the non-infarcted myocardium adjacent to the aneurysm (borderzone) and in myocardium remote from the aneurysm were estimated by minimizing the errors between FE model-predicted and measured systolic strains and LV volumes using the successive response surface method for optimization. The significant depression in optimized T(max_B) relative to T(max_R) was confirmed by direct ex vivo force measurements from skinned fiber preparations. The optimized values of T(max_B) and T(max_R) were not overly sensitive to the passive material parameters specified. The computation time of less than 5 hours associated with our proposed method for estimating regional myocardial contractility in vivo makes it a potentially very useful clinical tool.

Response surface methodology for the rapid design of aluminum sheet metal forming parameters

We focus on a successive response surface method for the optimization problems. The response surfaces are built using Moving Least Squares approximations constructed within a moving region of interest. Our first approach is an extension of pattern search algorithms with a fixed pattern panned and zoomed in a continuous manner across the design space. In the second one, the region of interest moves across a predefined discrete grid of authorized experimental designs. Two applications of the sheet metal forming process are used to demonstrate the robustness of the method. We use the one-step Inverse Approach as a surrogate model during the optimization. The final design is validated with Stampack ®, Ls-Dyna ® and Abaqus ® commercial codes based respectively on explicit dynamic and implicit static approaches.

Inverse Parameter Fitting of Biological Tissues: A Response Surface Approach

In this paper, we present the application of a semi-global inverse method for determining material parameters of biological tissues. The approach is based on the successive response surface method, and is illustrated by fitting constitutive parameters to two nonlinear anisotropic constitutive equations, one for aortic sinus and aortic wall, the other for aortic valve tissue. Material test data for the aortic sinus consisted of two independent orthogonal uniaxial tests. Material test data for the aortic valve was obtained from a dynamic inflation test. In each case, a numerical simulation of the experiment was performed and predictions were compared to the real data. For the uniaxial test simulation, the experimental targets were force at a measured displacement. For the inflation test, the experimental targets were the three-dimensional coordinates of material markers at a given pressure. For both sets of tissues, predictions with converged parameters showed excellent agreement with the data, and we found that the method was able to consistently identify model parameters. We believe the method will find wide application in biomedical material characterization and in diagnostic imaging.

Moving least squares response surface approximation: Formulation and metal forming applications

We focus on a successive response surface method for the optimization problems. The response surfaces are built using Moving Least Squares approximations constructed within a moving region of interest. Our first approach is an extension of pattern search algorithms with a fixed pattern panned and zoomed in a continuous manner across the design space. In the second one, the region of interest moves across a predefined discrete grid of authorized experimental designs. Two examples of the sheet metal forming process are used to demonstrate the robustness of the method. We use the one-step Inverse Approach as a surrogate model during the optimization. The final designs is validated with Stampack commercial code based on Explicit Dynamics Approach.

Material identification in structural optimization using response surfaces

This paper evaluates the use of a response surface optimization algorithm for structural material or parameter identification. The algorithm used is the successive response surface method (SRSM) as implemented in LS-OPT. Two methods are used in the formulation of the optimization problem. The first is to minimize the maximum deviation of the distance function between the simulated and experimental results at selected points, while the second approach minimizes the more standard least squares residual form of the distance function, effectively providing a compromised match over all the parameters selected. SRSM uses a trust region that is adapted using a heuristic contraction and panning approach. The method has only one user-required parameter, the size of the initial trust region. To illustrate the robustness of SRSM as a material identification tool, three test cases are presented. The first concerns the identification of the power-law material parameters of a simple tensile test specimen. The second test case determines the leakage coefficient-pressure load curve of an airbag given experimental kinematic data of a chest form impacting the airbag. In the third test case the material identification of a rate-dependent low-density foam material is conducted. It is shown that SRSM essentially converges within 10 iterations for all the test cases, and that the two distance function minimization approaches produce similar results.

An improved version of DYNAMIC‐Q for simulation‐based optimization using response surface gradients and an adaptive trust region

AbstractThis paper proposes an improvement on the DYNAMIC‐Q method for optimization of applications requiring expensive and (often) noisy numerical simulations for function evaluations. The original DYNAMIC‐Q method requires the selection of a step size and a move limit by the user. For noisy responses, the success of the original method depends on the correct choice of step size and move limit. The improved version uses the adaptive trust region features of an existing successive response surface method (SRSM) algorithm to replace the selection of the step size and move limit with one parameter, the initial trust region size. For the testing of the new method, DYNAMIC‐Q(RSG), an arbitrary selection from the well‐known Hock and Schittkowski problems as well as a non‐linear structural engineering problem are used. It is shown that for problems where the gradient information is reliable, the original DYNAMIC‐Q method is superior as expected. However, where noisy responses cause inaccurate and unreliable gradient information, the SRSM and DYNAMIC‐Q(RSG) methods are proven to be more robust with SRSM superior in its ability to converge more stably. The performance of DYNAMIC‐Q(RSG) is encouraging as it maintains the robustness and minimum user input of SRSM, but has faster convergence for analytical problems. Copyright © 2003 John Wiley & Sons, Ltd

On the robustness of a simple domain reduction scheme for simulation‐based optimization

This paper evaluates a Successive Response Surface Method (SRSM) specifically developed for simulation‐based design optimization, e.g. that of explicit nonlinear dynamics in crashworthiness design. Linear response surfaces are constructed in a subregion of the design space using a design of experiments approach with a D‐optimal experimental design. To converge to an optimum, a domain reduction scheme is utilized. The scheme requires only one user‐defined parameter, namely the size of the initial subregion. During optimization, the size of this region is adapted using a move reversal criterion to counter oscillation and a move distance criterion to gauge accuracy. To test its robustness, the results using the method are compared to SQP results of a selection of the well‐known Hock and Schittkowski problems. Although convergence to a small tolerance is slow when compared to SQP, the SRSM method does remarkably well for these sometimes pathological analytical problems. The second test concerns three engineering problems sampled from the nonlinear structural dynamics field to investigate the method's handling of numerical noise and non‐linearity. It is shown that, despite its simplicity, the SRSM method converges stably and is relatively insensitive to its only user‐required input parameter.