Nielen Stander

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 methods 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.

Automotive crashworthiness design using response surface-based variable screening and optimization

Purpose – The purpose of this paper is to provide a methodology with which to perform variable screening and optimization in automotive crashworthiness design.Design/methodology/approach – The screening method is based on response surface methodology in which linear response surfaces are used to create approximations to the design response. The response surfaces are used to estimate the sensitivities of the responses with respect to the design variables while the variance is used to estimate the confidence interval of the regression coefficients. The sampling is based on the D‐optimality criterion with over‐sampling to improve noise filtering and find the best estimate of the regression coefficients. The coefficients and their confidence intervals as determined using analysis of variance (ANOVA), are used to construct bar charts for the purpose of selecting the important variables.Findings – A known analytical function is first used to illustrate the effectiveness of screening. Using the finite element me...

A dynamic penalty function method for the solution of structural optimization problems

Abstract This paper presents an adaptation of an existing dynamic trajectory method for unconstrained minimization to handle constrained optimization problems. This is done by the application of a dynamic penalty parameter procedure to allow for the constraints. The method is applied to structural optimization problems that involve the determination of minimum weight structures of trusses and frames, subject to stress, displacement, and frequency constraints, under various prescribed load conditions. Because structural problems, in general, require detailed finite-element analyses to evaluate the constraint functions, the direct application of the trajectory method, requiring updated information at each step along the path, would be expensive. This problem is overcome by the successive application of the trajectory method to approximate quadratic subproblems that can be solved economically. The comprehensive new approach is called the DYNAMIC-Q method. The method is successfully applied to a number of truss and frame problems and is found to be both reliable and easy to use.

OPTIMIZATION OF A SHEET METAL FORMING PROCESS USING SUCCESSIVE MULTIPOINT APPROXIMATIONS

An automated optimization method based on multipoint approximations and applied to the design of a sheet metal forming process is presented. Due to the highly complex nature of the design functions, it was decided to focus on the characterization of the final product thickness distribution as a function of the preforming die shape variables. This was achieved by constructing linear approximations to the noisy responses usingresponse surface methodology (RSM). These approximations are used to obtain an approximate solution to an optimization problem. Successive approximations are constructed, which improve the solution. An automated panning-zooming scheme is used to resize and position the successive regions of approximation. The methodology is applied to optimally design the preforming die shape used in the manufacture of an automotive wheel centre pressing from a sheet metal blank. The die shape is based on a cubic spline interpolation and the objective is to minimize the blank weight, subject to minimum thickness constraints. A weight saving of up to 9.4% could be realized for four shape variables. Restart is introduced to escape local minima due to the presence of noise and to accelerate the progress of the optimization process.

A Comparison of Metamodeling Techniques for Crashworthiness Optimization

This crashworthiness optimization study compares the use of three metamodeling techniques while using a sequential random search method as a control procedure. The three methods currently applied are (i) the Successive Linear Response Surface Method, (ii) the Updated Neural Network method and (iii) the Kriging method. Three crashworthiness examples, including a full vehicle multidisciplinary analysis, are investigated. It is shown that, although NN and Kriging seem to require a larger number of initial points, the three metamodeling methods have comparable efficiency when attempting to achieve a converged result. The Neural Network and Kriging methods have the advantage that they can be updated to construct a reasonable global approximation with higher accuracy at the optimum.

An efficient 4‐node 24 D.O.F. thick shell finite element with 5‐point quadrature

A 4‐node flat shell quadrilateral finite element with 6 degrees of freedom per node, denoted QC5D‐SA, is presented. The element is an assembly of a modification of the drilling degree of freedom membrane presented by Ibrahimbegovic et al., and the assumed strain plate element presented by Bathe and Dvorkin. The part of the stiffness matrix associated with in—plane displacements and rotations is integrated over the element domain by a modified 5‐point reduced integration scheme, resulting in greater efficiency without the sacrifice of rank sufficiency. The scheme produces a soft higher order deformation mode which increases numerical accuracy. A large number of standard benchmark problems are analyzed. Some examples show that the effectiveness of a previously proposed “membrane locking correction” technique is significantly reduced when employing distorted elements. However, the element is shown to be generally accurate and in many cases superior to existing elements.

The genetic algorithm applied to stiffness maximization of laminated plates: review and comparison

The design of laminated structures is highly tailorable owing to the large number of available design variables, thereby requiring an optimization method for effective design. Furthermore, in practice, the design problem translates to a discrete global optimization problem which requires a robust optimization method such as the genetic algorithm. In this paper, the genetic algorithm, based on the real variable coding, is applied to the strain energy minimization of rectangular laminated composite plates. The results for both a point load and uniformly distributed load compare well with those achieved using trajectory methods for continuous global optimization.

A pseudo-discrete rounding method for structural optimization

A new heuristic method aimed at efficiently solving the mixed-discrete nonlinear programming (MDNLP) problem in structural optimization, and denotedselective dynamic rounding, is presented. The method is based on the sequential rounding of a continuous solution and is in its current form used for the optimal discrete sizing design of truss structures. A simple criterion based on discrete variable proximity is proposed for selecting the sequence in which variables are to be rounded, and allowance is made for both upward and downward rounding. While efficient in terms of the required number of function evaluations, the method is also effective in obtaining a low discrete approximation to the global optimum. Numerical results are presented to illustrate the effectiveness and efficiency of the method.

Optimal discrete sizing of truss structures subject to buckling constraints

The selective dynamic rounding (SDR) algorithm previously developed by the authors, and based on a dual step rounding approach, is used for the optimal sizing design of truss structures subject to linear buckling constraints. The algorithm begins with a continuous optimum followed by a progressive freezing of individual variables while solving the remaining continuous problems. The allowable member stresses are predicted by the linear regression of the tabular section properties, while the exact allowable compressive stresses are back-substituted for those variables fixed on discrete values in each intermediate mixed-discrete nonlinear problem. It is shown that a continuous design based on the regression analysis of section effectiveness vs. area is effective as a starting point for the dual step discrete optimization phase. A range of examples is used to illustrate that with “conservative” regression, discrete designs can be achieved which are significantly lighter than those in which the variables have been rounded up.

Thermal optimization in transient thermoelasticity using response surface approximations

Response surface methodology is used to construct approximations to temperature and stress in transient thermoelastic analysis of non-linear systems. The analysis forms the core component of a heating/cooling rate maximization problem in which the ordinates of the ambient temperature at equally spaced time intervals are chosen as the design variables. Polynomials or cubic splines are fitted through the ordinates to describe the ambient temperature profile required for the convective heat transfer analysis. An experimental design method based on D-optimality and a genetic algorithm was used to select the design points used to create the approximations. Linear response surfaces were found to be sufficiently accurate, thereby minimizing the number of finite element analyses. Two examples of which one is a thick-walled pressure vessel are used to illustrate the methodology.