In this paper, space mapping (SM) technique is combined with response surface methodology (RSM). SM is an optimization method well suited for very costly problems to find an improved design with fulfilled constraints. The SM technique use less costly models, which complements the correct models. The theory is established and compared to the corrected RSM. A multipoint version of SM is presented, where a separate evaluation is done in each iteration to improve the mapping function. Using this additional evaluation to update the mapping function, generally, the number of iterations to find the optimum solution can be reduced. Thus, the elapsed time to solve the optimization problem can be reduced if a parallel computer is utilized. Finally, one engineering optimization problem is solved to illustrate the application of SM in vehicle crashworthiness structural optimization.

In this paper the Response Surface Methodology (RSM) and the Stochastic Optimization (SO) are compared with regard to their efficiency and applicability in crashworthiness design. Optimization of simple analytic expressions and optimization of a front rail structure are application used in order to assess the respective qualities of both methods. A low detailed vehicle structure is optimized to demonstrate the applicability of the methods in engineering practice. The investigations reveal that RSM is favoured compared to SO for less than 10-15 design variables. A novel zooming method is proposed for SO, which improve its convergence behaviour. A combination of both the RSM and the SO is efficient. Stochastic Optimization can be used in order to determine an appropriate starting design for an RSM optimization, which continues the optimization. Two examples are investigated using this combined method.

In this paper the response surface methodology (RSM) and stochastic optimization (SO) are compared with regard to their efficiency and applicability in crashworthiness design. Optimization of simple analytic expressions and optimization of a front rail structure are the applications used to assess the respective qualities of both methods. A low detail vehicle structure is optimized to demonstrate the applicability of the methods in engineering practice. The investigations reveal that RSM is better compared to SO for fewer than 10–15 design variables. The convergence behaviour of SO improves compared to RSM when the number of design variables is increased. A novel zooming method is proposed which improves the convergence behaviour. A combination of both the RSM and the SO is efficient, stochastic optimization could be used in order to determine appropriate starting points for an RSM optimization, which continues the optimization. Two examples are investigated using this combined method.

This dissertation addresses the problems and possibilities of using structural optimization in vehicle crashworthiness design. The first part of the thesis gives an introduction to vehicle crashworthiness design. The optimization methods presented are also used to exemplify how structural optimization and robustness analysis can be used in vehicle crashworthiness design.

In the second part of the thesis, five papers are appended, where different optimization methods are evaluated and improved for the usage in vehicle crashworthiness design. These papers concern the optimization methods Response Surface Methodology (RSM), Stochastic Optimization (SO) and Space Mapping (SM).

Each method has its advantages and disadvantages. The Response Surface Methodology is the easiest method to use and the method that most often finds the best design of these three methods. Generally RSM is rather expensive, especially when many design variables are used. Then, SO is an effective alternative because in this method the number of evaluations is independent of the number of design variables, which is not the case for RSM. Space Mapping is the cheapest method, because it needs only one or two evaluations per iteration. However, SM is generally a method to fmd an improved design with fulfilled constraints and sometimes not the absolute optimum solution but to a low cost. Hence, both RSM and SO may produce better designs but at the price of more response evaluations.

The aim of this work is to illustrate how a space mapping technique using surrogate models together with response surfaces can be used for structural optimization of crashworthiness problems. To determine the response surfaces, several functional evaluations must be performed and each evaluation can be computationally demanding. The space mapping technique uses surrogate models, i.e. less costly models, to determine these surfaces and their associated gradients. The full model is used to correct the gradients from the surrogate model for the next iteration. Thus, the space mapping technique makes it possible to reduce the total computing time needed to find the optimal solution. First, two analytical functions and one analytical structural optimization problem are presented to exemplify the idea of space mapping and to compare the efficiency of space mapping to traditional response surface optimization. Secondly, a sub-model of a complete vehicle finite element (FE) model is used to study different objective functions in vehicle crashworthiness optimization. Finally, the space mapping technique is applied to a structural optimization problem of a large industrial FE vehicle model, consisting of 350.000 shell elements and a computing time of 100 h. In this problem the intrusion in the passenger compartment area was reduced by 32% without compromising other crashworthiness parameters.

This paper presents a method where a multi objective optimization techniqueis used together with response surface methods in order to support crashworthiness design. As in most engineering design problems thereare several conflicting objectives that have to be considered when formulating a design problem as an optimization problem. Here this is exemplified by the desire to minimize the intrusion into the passenger compartment area and simultaneously obtain low maximum acceleration during vehicle impact. These two objectives are naturally conflicting, since low maximum acceleration implies large intrusion. The contribution of thispaper is to show a successful application of a set of existing methods to solve a real world engineering problem.The paper also presents methods of illustrating the results obtained from the multi-objective optimization.

The aim of this paper is to determine if the Space Mapping technique using surrogate models together with response surfaces is useful in the optimization of crashworthiness and sheet metal forming. In addition, the efficiency of optimization using Space Mapping will be compared to traditional structural optimization using the Response Surface Methodology (RSM). Five examples are used to study the algorithm: one optimization of an analytic function and four structural optimization problems. All examples are constrained optimization problems. In all examples, the algorithm converged to an improved design with all constraints fulfilled, even when a conventional RSM optimization failed to converge. For the crashworthiness design problems, the total computing time for convergence was reduced by 53% using Space Mapping compared to conventional RSM. For the sheet metal forming problems the total computing time was reduced by 63%. The conclusions are that optimization using Space Mapping and surrogate models can be used for optimization in crashworthiness design and sheet metal forming applications with a significant reduction in computing time.

The aim of this work is to determine when a linear surface approximation in the Response Surface Methodology can be utilised to calculate parameter sensitivities of a vehicle exposed to impact loading. Linear surface approximations are less costly than quadratic surfaces due to a lower number of function evaluations. However, the linear surface approximation error is often larger. Two structural impact problems are studied: one simple rail structure with two design variables and one vehicle model with ten design variables. The study has shown that linear surface approximations can be used to determine the order of sensitivity. However, the absolute magnitudes of the sensitivities are not reliable. This can be used if the normalised root mean square error is smaller than 10%.

The aim of this work is to illustrate how Space Mapping technique using surrogate models together with response surfaces can be used for structural optimization of crashworthiness problems. To determine the response surfaces, several functional evaluations must be performed and each evaluation can be computationally demanding. Space Mapping technique uses surrogate models, i.e. less costly models, to determine these surfaces and their associated gradients. The full model is used to correct the gradients from the surrogate model for the next iteration. Thus, Space Mapping technique makes it possible to reduce the total computing time needed to find the optimal solution. First, one simple structural optimization problem is presented to exemplify the idea of Space Mapping and to compare the efficiency of Space Mapping to traditional response surface optimization. Secondly, a sub-model of a complete vehicle FE model is used to study different objective functions in vehicle crashworthiness optimization. Finally, the Space Mapping technique is applied to a structural optimization problem of a complete FE vehicle model, consisting of 350.000 shell elements and a computing time of 100 h. In this problem the intrustion in the passenger compartment area was reduced by 16% without compromising other crashworthiness parameters.

The aim of this paper is to determine the efficient number of experimental points when using the response surface methodology in crashworthiness problems.

The D-optimality criterion is used as experimental design method. Two application models have been studied, one square tube and one front rail from Saab Automobile AB. Both models were fully parameterized in the preprocessor LS-INGRID but only two design variables were used. The optimization package LS-OPT was used to determine the design of experiments using the D-optimality criterion. Both models were subjected to an impact into a rigid wall and the simulations were carried out using LS-DYNA. A general recommendation is to to use 1.5 times the minimum number of experimental points. A more specialized recommendation is for linear surfaces 1.5, elliptic surfaces 2.2 and for quadratic surfaces 1.6 times the minimum number of experimental points.