It is important to consider variations in material parameters in the design of automotive structures in order to obtain a robust and reliable design. However, expensive tests are required to gain complete knowledge of the material behavior and its associated variation. Consequently, due to time and cost constraints, simplified material scatter modeling techniques based on scatter data of typical material properties provided by the material suppliers are used at early design stages in simulation-based robustness studies. The aim of this paper is to study the accuracy of the simplified scatter modeling methods in representing the real material variation. The simplified scatter modeling methods are evaluated by comparing the material scatter obtained by them to the scatter obtained by complete tensile tests, which are obtained after detailed timeconsuming experimental investigations. Furthermore, an accuracy assessment is carried out based on selected responses from an axially-crushed, square tube made from DP600 steel.

Multidisciplinary design optimisation (MDO) can be used as an effective tool to improve the design of automotive structures. Large-scale MDO problems typically involve several groups who must work concurrently and autonomously in order to make the solution process efficient. In this article, the formulations of existing MDO methods are compared and their suitability is assessed in relation to the characteristics of automotive structural applications. Both multi-level and single-level optimisation methods are considered. Multi-level optimisation methods distribute the design process but are complex. When optimising automotive structures, metamodels are often required to relieve the computational burden of detailed simulation models. The metamodels can be created by individual groups prior to the optimisation process, and thus offer a way of distributing work. Therefore, it is concluded that a single-level method in combination with meta-models is the most straightforward way of implementing MDO into the development of automotive structures. If the benefits of multi-level optimisation methods, in a special case, are considered to compensate for their drawbacks, analytical target cascading has a number of advantages over collaborative optimisation, but both methods are possible choices.

It is essential to account for variations in the manufacturing process and in loading conditions when improving the robustness and reliability of a product’s design. A finite element study of the robustness of a hat profile beam made from boron steel subjected to a three point bending load is presented, and an approach to incorporate the variations investigated is demonstrated. Fracture risk factors and the maximum deflection of the beam are the measured responses. Spatial variation of the sheet thickness is considered in the forming simulations, along with other input variations. Stress-strain relations from tensile tests have been used in the robustness analyses to represent the variation in material properties. Furthermore, validations of four metamodels have been performed. Both the responses measured were found to be sensitive to input variations. Separate metamodels were created for each risk prone zone in order to improve the performance of the metamodels for risk factor responses.

Spot welds are commonly used to join steel sheets in automotive structures. The number and layout of these spot welds are vital for the performance of the structure. However, reducing the number of spot welds will cut both production time and cost. This article presents three different methods of reducing the number of spot welds in automotive structures: ranking-based selection, topology optimization and size optimization of a parameterized model. The methods are compared in a simple example and it is found that the latter two methods have the best potential of reducing the number of spot welds. Topology optimization requires less preparation and computational effort as compared to size optimization of a parameterized model. However, the method is primarily suitable for studies where load cases involving linear systems are judged to be most important. Otherwise, size optimization of a parameterized model is probably a better choice. The topology optimization approach is successfully demonstrated in a full-scale industrial application example and confirms that the method is useful within contemporary product development.

Automotive companies continuously strive to design better products faster and more cheaply using simulation models to evaluate every possible aspect of the product. Multidisciplinary design optimization (MDO) can be used to find the best possible design taking into account several disciplines simultaneously, but it is not yet fully integrated within automotive product development. The challenge is to find methods that fit company organizations and that can be effectively integrated into the product development process. Based on the characteristics of typical automotive structural MDO problems, a metamodel-based MDO process intended for large-scale applications with computationally expensive simulation models is presented and demonstrated in an example. The process is flexible and can easily fit into existing organizations and product development processes where different groups work in parallel. The method is proven to be efficient for the discussed example and improved designs can also be obtained for more complex industrial cases with comparable characteristics.

Optimisation of vehicle design is necessary to meet increased safety requirements, new emission regulations, and to deal with competition in the global market, etc. However, optimised design using classical optimisation techniques with deterministic models might not meet the desired performance level or might fail in extreme events in real life owing to uncertainties in the design parameters and loading conditions. Consequently, it is essential to account for uncertainties in a systematic manner to generate a robust and reliable design. In this paper, an approach to perform multiobjective, reliability-based, and robust design optimisation is presented using a vehicle side impact crashworthiness application. Metamodels have been used in the optimisation process to decrease computational effort. Variations in material properties, thicknesses, loading conditions, and B-pillar heat-affected zone material strength have been considered for the stochastic optimisation. A comparative study of deterministic, reliability-based, and robust optimisation approaches is performed.

Mixed isotropic-distortional hardening allows for individual stress-plastic strain relations in different straining directions. Such hardening can be obtained by allowing the parameters in the effective stress function depend on anisotropy functions of the equivalent plastic strain. A methodology to calibrate these anisotropy functions is proposed in this work, and is demonstrated on an austenitic strainless steel. A high exponent eight parameter effective stress function for plane stress states is utilised. The anisotropy functions are calibrated by the use of experimental data from uniaxial tensile test data in three material directions and a balanced biaxial test. The plastic anisotropy is evaluated at a finite number of plastic strains, and it is assumed to vary piecewise linearly with respect to the equivalent plastic strain. At each level of plastic strain, the anisotropy is correctly represented, even if rather large increments in plastic strain are used in the calibration. It was found that there are at least two sets of anisotropy functions which satisfy the conditions in the calibration procedure. The resulting uniaxial stress-strain relations from the two sets of anisotropy functions in four additional straining directions, not included in the calibration set, were compared to the corresponding experimental data. From this validation, one of the anisotropy function sets could be discarded, whereas the other one gave a good prediction of the stress-strain relations in all the four additional directions. (C) 2015 Elsevier Ltd. All rights reserved.

It is important to understand the strain recovery of a steel sheet in order to predict its springback behaviour. During strain recovery, the stress–strain relation is non-linear and the resulting unloading modulus is decreased. Moreover, the unloading modulus will degrade with increasing plastic pre-straining. This study aims at adding new knowledge on these phenomena and the mechanisms causing them. The unloading behaviour of the dual-phase steel DP600 is characterised experimentally and finite element (FE) simulations of a representative volume element (RVE) of the microstructure are performed. The initial stress and strain state of the micromechanical FE model is found by a simplified simulation of the annealing processes. It is observed from the experimental characterisation that the decrease of the initial stiffness of the unloading is the main reason for the degrading unloading modulus. Furthermore, the developed micromechanical FE model exhibits non-linear strain recovery due to local plasticity caused by interaction between the two phases.

The main objective of this work is to examine the use of anisotropic fracture criteria in order to predict fracture in dual-phase steel. The introduction of a material directional function into the fracture criterion was used in order to account for anisotropy observed in experiments. Selected fracture criteria were fist calibrated by ordinary tensile and in-plane shear tests using specimens cut in three material directions. In order to validate the performance, two types of validation tests were conducted. First, plane strain (notched tensile) tests were carried out in three material directions. Second, Nakajima tests with a waist of 130 mm were conducted, also in three material directions. The fit to the calibration tests was improved with all material directional functions compared to the isotropic criterion. Overall best performance was achieved when a material direction function based on the structural tensors was introduced.

This paper presents a study of two different hybrid composite-aluminum concepts applied to a winglike structure which is exposed to mechanical  and thermal load. The aim of the study is to determine the most suitable  hybrid concept to later on be used in structural fatigue and static testing. In both concepts, the mass is optimized with respect to two different sets of requirements, one of which is currently in use in the fighter aircraft industry and one which is a modified version of the current requirement set. The issues considered in the study are mass, thermal behavior, buckling, bolted joints, failure criteria and fatigue damage, and they are examined in the frame of both requirement sets. The results clearly indicate the order of criticality between the different criteria in the different parts of each concept. Also, the comparison of two requirement sets gives an idea of the degree of influence of the modified criteria on the hybrid concepts and their mass. Based on the mass and the structural behavior in a thermal-mechanical loading one of the hybrid concepts is chosen for further studies and testing.

Failure in ductile sheet metal structures is usually caused by one, or a combination of, ductile tensile fractures, ductile shear fractures or localised instability. In this paper the failure characteristics of the high strength steel Docol 600DP are explored. The study includes both experimental and numerical sections. In the experimental sections, the fracture surface of the sheet subjected to Nakajima tests is studied under the microscope with the aim of finding which failure mechanism causes the fracture. In the numerical sections, finite element (FE) simulations have been conducted using solid elements. From these simulations, local stresses and strains have been extracted and analysed with the aim of identifying the fracture dependency of the stress triaxiality and Lode parameter.

A three-dimensional, solid finite element model of a composite-aluminum single-lap bolted joint with a countersunk titanium fastener is developed. The model includes progressive damage behavior of the composite and a plasticity model for the metals. The response to static loading is compared to experimental results from the literature. It is shown that the model predicts the initiation and the development of the damage well, up to failure load. The model is used to evaluate the local force-displacement responses of a number of single-lap joints installed in a hybrid composite-aluminum wing-like structure. A structural model is made where the fasteners are represented by two-node connector elements which are assigned the force-displacement characteristics determined by local models. The behavior of the wing box is simulated for bending and twisting loads applied together with an increased temperature and the distribution of fastener forces and the progressive fastener failure is studied. It is shown that the fastener forces caused by the temperature difference are of significant magnitude and should be taken into account in the design of hybrid aircraft structures. It is concluded that, the account of the non-linear response of the joints results in a less conservative load distribution at ultimate failure load.

In this work, selected fracture criteria are applied to predict the fracture of dualphase steel subjected to non-linear strain paths. Furthermore, the effects of manufacturing history are studied. Four fracture criteria were calibrated in three tests using standard specimens. The fracture criteria were first validated in the circular Nakajima test. A second validation test case was included in order to validate fracture prediction for non-linear strain paths. In this test a sheet metal component was manufactured and subsequently stretched until it fractured. All fracture criteria included in this study predict fracture during the Nakajima test with reasonable accuracy. In the second validation test however, the different fracture criteria show considerable diversity in accumulated damage during manufacturing which caused substantial scatter of the fracture prediction in the subsequent stretching. This shows that manufacturing history influences the prediction of fracture.

In order to accurately predict the mechanical properties of a sheet metal assembly it has been shown important to account for how the geometry and material properties are affected by the manufacturing process. It is also of a great interest to predict the variations of important responses, and how these variations depend on the manufacturing process.

In this study, the variation of properties during the multi-stage manufacturing process of a sheet metal assembly is evaluated and the variability of a response due to loading is studied. A methodology to investigate how variations evolve during the assembling process is presented. The multi-stage assembling process is virtually segmented, such that stochastic analyses of each process stage are performed and coupled to succeeding stages in order to predict the variation in properties of the final assembly. The methodology is applied to an industrial assembly and experimental validations have been conducted. The prediction of the geometry of the final assembly is in good agreement with the experimental results, while the prediction of the variation of this geometry is in fair agreement.

As demand for faster product development increases, physical prototypes are replaced by virtual prototypes. By using finite element simulations to evaluate the functional behaviour of the product as well as its manufacturing process, more design alternatives can be evaluated while a considerably smaller number of physical prototypes are needed. As sheet metal assemblies are common in a wide range of products, reliable methods for predicting their properties are necessary. By sequentially simulating the complete manufacturing process chain of an assembly, early predictions concerning the geometry and material properties of the assembly can be made.

In this study a simulation-based sensitivity study is performed in order to investigate the influence of the forming history on the predictions of assembly properties. In the study, several simulations of the assembly stage are performed in which different types of forming histories are retained from the forming stage. The simulations of the assembly stage will range from a case with linear elastic conditions without forming history, to a case with the full forming history state and consistent material modelling throughout all simulations. It is found that the residual stress state is the most influential history variable from the forming stage. Especially for more complex geometries in which large residual stresses can be retained.

Failure in sheet metal structures of ductile material is usually caused by one of, or a combination of, ductile fracture, shear fracture or localised instability. In this paper the failure of the high strength steel Docol 600DP and the ultra high strength steel Docol 1200M is explored. The constitutive model used in this study includes plastic anisotropy and mixed isotropic-kinematic hardening. For modelling of the ductile and shear fracture the models presented by Cockroft–Latham and Bressan–Williams have been used. The instability phenomenon is described by the constitutive law and the finite element (FE) models. For calibration of the failure models and validation of the results, an extensive experimental series has been conducted including shear tests, plane strain tests and Nakajima tests. The geometries of the Nakajima tests have been chosen so that the first quadrant of the forming limit diagram (FLD) were covered. The results are presented both in an FLD and using prediction of force–displacement response of the Nakajima test employing element erosion during the FE simulations. The classical approach for failure prediction is to compare the principal plastic strains obtained from FE simulations with experimental determined forming limit curves (FLCs). It is well known that the experimental FLC requires proportional strains to be useful. In this work failure criteria, both of the instability and fracture, are proposed which can be used also for non-proportional strain paths.

In the search for lightweight automobile designs, it is necessary to assure that robust crashworthiness performance is achieved. Structures that are optimized to handle a finite number of load cases may perform poorly when subjected to various dispersions. Thus, uncertainties must be accounted for in the optimization process. This article presents an approach to optimization where all design evaluations include an evaluation of the robustness. Metamodel approximations are applied both to the design space and the robustness evaluations, using artifical neural networks and polynomials, respectively. The features of the robust optimization approach are displayed in an analytical example, and further demonstrated in a large-scale design example of front side members of a car. Different optimization formulations are applied and it is shown that the proposed approach works well. It is also concluded that a robust optimization puts higher demands on the finite element model performance than normally.

Structural optimization is a valuable tool to improve the performance of products, but it is in general expensive to perform due to the required extensive number of function evaluations. Therefore, an approximate method based on the internal energy distribution, which only requires a small number of function evaluations, is presented here. By this method, structural optimization can be enabled already in the initial steps in the design of new products when fast, but not necessarily precise, results are often desired. However, the accuracy of the approximate solution depends on the structural behaviour. The internal energy based optimization method is here validated for three structures, but it is believed to be applicable to any structure subjected to a single load where the functions considered are related to the displacement of the loaded area and/or the material thicknesses of the structural parts.

In robustness studies, variations of material properties are often represented by simple assumptions, such as scaling of stress-strain relations, often due to lack of knowledge or deeper understanding of the material physics and the material model applied. By performing material characterisation tests on several batches of a DP600 steel and fitting a phenomenological material model to each batch, this paper studies the dispersion of material model parameters, as well as correlations between both experimental and model parameters. It is concluded that some of the charcterisation tests may be omitted in the future, due to correlations found between parameters. The results may also be applied in a robustness study by inversely using the retrieved statistics to generate reasonable new sets of material model parameters. The methodology presented may be adopted for any other type of material characterisation process.

This paper discusses important improvements in the efficient Critical Constraint Method (CCM) for the optimization of structural product families subjected to multiple crash load cases. The method was first presented by Öman and Nilsson (Struct Multidisc Optim 41(5):797–815, 2010). However, the algorithm often converged towards an infeasible solution, which considerably limited the applicability of the method. Therefore, improvements are presented here to make the method more robust regarding feasible solutions, resulting in only a minor decrease in efficiency compared to the original method. The improvements include; a penalty approach to control the feasibility of the method by continuously pushing the solution out of the infeasible region, a dynamic contraction algorithm to increase the accuracy and robustness of the method by considering the optimization progress and variable history in the reduction of the step size, and the implementation of a parallel approach to further increase the efficiency of the method by enabling the full potential of large-scale computer clusters. Finally, the potential of the improved CCM algorithm is demonstrated on a large-scale industrial family optimization problem and it is concluded that the high efficiency of the method enables the usage of large product family optimization in the design process.

An increasing number of components in automotive structures are today made from advanced high strength steel (AHSS). Since AHSS demonstrates more severe springback behaviour than ordinary mild steels, it requires more efforts to meet the design specification of the stamped parts. Consequently, the physical fine tuning of the die design and the stamping process can be time consuming. The trial-and-error development process may be shortened by replacing most of the physical try-outs with finite element (FE) simulations of the forming process, including the springback behaviour. Still it can be hard to identify when a stamped part will lead to an acceptable assembly with respect to the geometry and the residual stress state. In part since the assembling process itself will distort the components. To resolve this matter it is here proposed to extend the FE-simulation of the stamping process, to also include the first level sub-assembly stage. In this study a methodology of sequentially simulating each step in the manufacturing process of an assembly is proposed. Each step of the proposed methodology is described, and a validation of the prediction capabilities is performed by comparing with a physically manufactured assembly. The assembly is composed of three sheet metal components made from DP600 steel which are joined by spot welding. The components are designed to exhibit severe springback behaviour in order to put both the forming and subsequent assembling simulations to the test. The work presented here demonstrates that by using virtual prototyping it is possible to predict the final shape of an assembled structure.

When designing a complex product, many groups are concurrently developing different parts or aspects of the product using detailed simulation models. Multidisciplinary design optimization (MDO) has its roots within the aerospace industry and can effectively improve designs through simultaneously considering different aspects of the product. The groups involved in MDO need to work autonomously and in parallel, which influence the choice of MDO method. The methods can be divided into single-level methods that have a central optimizer making all design decisions, and multi-level methods that have a distributed decision process.

This report is a comprehensive summary of the field of MDO with special focus on structural optimization for automotive applications using metamodels. Metamodels are simplified models of the computationally expensive detailed simulation models and can be used to relieve some of the computational burden during MDO studies. The report covers metamodel-based design optimization including design of experiments, variable screening, metamodels and their validation, as well as optimization methods. It also includes descriptions of several MDO methods, along with a comparison between the aerospace and automotive industries and their applications of MDO.

The information in this report is based on an extensive literature survey, but the conclusions drawn are influenced by the authors’ own experiences from the automotive industry. The trend goes towards using advanced metamodels and global optimization methods for the considered applications. Further on, many of the MDO methods developed for the aerospace industry are unsuitable for the automotive industry where the disciplines are more loosely coupled. The expense of using multi-level optimization methods is then greater than the benefits, and the authors therefore recommend single-level methods for most automotive applications.

The plastic hardening of metastable austenitic stainless steel is partly governed by martensitic transformation, the occurrence of serrated plastic flow, and plastic strain ageing phenomena. In this paper an elasto-viscoplastic material model with isotropic distortional plastic hardening is developed. The model accounts for static and dynamic strain ageing as well as the martensitic transformation. An experimental programme has been conducted in order to fit the model parameters to an austenitic stainless steel within the EN 1.4310 standard. The identification of the dynamic strain ageing was based on so called jump tests, where a sudden strain rate increase was shown to result in an instantaneous positive strain rate sensitivity followed by negative steady state strain rate sensitivity. Furthermore, the static strain ageing was identified by unloading tensile test specimens at specified plastic strains and then reloading these specimens after different periods of time. The observed material behaviour in the test situations can be predicted by the developed model. Lastly, the model was validated by predicting the force-displacement relation of the material in a shear test: the prediction agrees well with experimental results.

This paper presents an evaluation of the constitutive behaviour, including plastic anisotropy and mixed isotropic-kinematic hardening of two high strength steels, Docol 600DP and Docol 1200M, during strain path changes. A series of tensile and shear tests was performed on both virgin and pre-strained materials. The initial anisotropy and work hardening parameters were obtained from tensile tests, shear tests and a bulge test of the virgin material, whereas the kinematic hardening parameters were identified by comparing numerical predictions to experimental results related to the pre-strained materials. Numerical predictions using the obtained parameters agree well with the experimental results, both in the case of proportional, and under non-proportional strain paths.

Robustness evaluations give valuable information about the sensitivity of a structure to stochastic variation of design parameters, but are in general expensive to perform due to the extensive number of function evaluations required. Therefore, an Internal Energy Based (IEB) method is presented here, in which the structural sensitivity to variation of each thickness parameter is estimated based on the internal energy distribution obtained from one single function evaluation. In this way, the cost of such structural robustness evaluations is significantly reduced. However, the accuracy of the IEB method depends on the structural behaviour. Here it is tested for two structures with nonlinear displacement responses caused by impact loads. The method is however believed to be applicable to any structure exposed to a single load acting on a limited area of the structure, and for response functions related to the displacement of the loaded area.

A product of high quality is a product that performs well, not only in exactly the situations it was designed to handle but also in slightly different situations that arise in the usage of the product. As a specific example, the performance of a bumper system should not depend on small fluctuations in the manufacturing process or on small variations in the impact event. In this work, the robustness of an existing vehicle bumper system subjected to a crash load has been evaluated both experimentally and numerically. In the latter case, different widely used approaches to numerically assess the robustness have been utilised. A reliable numerical robustness study provides the designer with a valuable tool for improving a design, and an evaluation of these methods in this context is therefore of interest. It is concluded that for the example under study, both the Monte Carlo method and the metamodel-based Monte Carlo methods work well. Furthermore, for moderate dispersions levels, i.e. a small design space with no bifurcation in the deformation pattern, a linear response approximation is shown to have a sufficient accuracy to be used in the metamodel-based robustness analysis. The performed numerical robustness studies also point out that the performance of a nominal simulation, i.e. a simulation conducted with mean values for all variables, does not in general predict the mean performance of the finite element model. Finally, some possible design improvements for the bumper system under study are also identified

Structural optimization is a valuable tool to improve the performance of products, but it is in general expensive to perform due to the required extensive number of function evaluations. Therefore, an approximate method based on the internal energy distribution, which only requires a small number of function evaluations, is presented here. By this method, structural optimization can be enabled already in the initial steps in the design of new products when fast, but not necessarily precise, results are often desired. However, the accuracy of the approximate solution depends on the structural behaviour. The internal energy based optimization method is here validated for three structures, but it is believed to be applicable to any structure subjected to a single load where the functions considered are related to the displacement of the loaded area and/or the material thicknesses of the structural parts.

Conventionally optimized structures may show a tendency to be sensitive to variations, for instance in geometry and loading conditions. To avoid this, research has been carried out in the field of robust optimization where variations are taken into account in the optimization process. The overall objective is to create solutions that are optimal both in the sense of mean performance and minimum variability. This work presents an alternative approach to robust optimization, where the robustness of each design is assessed through multiple sampling of the stochastic variables at each design point. Meta-models for the robust optimization are created for both the mean value and the standard deviation of the response. Furthermore, the method is demonstrated on an analytical example and an example of an aluminium extrusion with quadratic cross-section subjected to axial crushing. It works well for the chosen examples and it is concluded that the method is especially well suited for problems with a large number of random variables, since the computational cost is essentially independent of the number of random variables. In addition, the presented approach makes it possible to take into consideration variations that cannot be described with a variable. This is demonstrated in this work by random geometrical perturbations described with the use of Gaussian random fields.

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This paper discusses the problem of structural optimization of product families subjected to multiple load cases, evaluated by computationally costly finite element analysis. Product families generally have a complex composition of shared components that makes individual product optimization difficult as the relation between the shared variables is not always intuitive. More optimal is to treat the problem as a product family optimization problem. Though, for product families subjected to multiple and computationally costly crash loads, the optimization problem takes too long time to solve with traditional methods. Therefore, a new optimization algorithm is presented that decomposes the family problem into sub-problems and iteratively reduces the number of sub-problems, decouple and solve them. The algorithm is applicable for module based product families with predefined composition of generalized commonality, subjected to multiple load cases that can be analyzed separately. The problem reduction is performed by only considering the constraints that are critical in the optimal solution. Therefore the optimization algorithm is called the Critical Constraint Method, CCM. Finally the CCM algorithm is evaluated by two product family optimization problems.

Optimised designs have a tendency of being sensitive to variations. It is therefore of great importance to analyse this sensitivity to assure that a design is robust, i.e. sufficiently insensitive to variations. To analyse robustness, variations are introduced in model parameters and their influences on simulation responses are studied. This is usually achieved using the Monte Carlo method. Though, due to the large number of simulations needed, the Monte Carlo method is very costly for problems requiring a long computing time. Therefore, in this work a meta model-based Monte Carlo method is used to evaluate the robustness of a vehicle structure. That is, the Monte Carlo analysis is performed on a surface approximation of the true response, over the domain of interest. The methodology used is to first identify the variables that influence the response the most, referred to as a screening, using simple linear response surfaces. This is followed by a more detailed sensitivity analysis using only the identified variables and a quadratic response surface, thereby incorporating second order effects. A truck cab model exposed to a pendulum impact load is used as an evaluation of this method, and the important variables and their influence on the response are identified. The effect of including results from forming simulations is also evaluated using the truck cab model. Variations are introduced before forming simulations, thereby taking forming effects into account in the sensitivity analysis. The method was found to be a good tool to identify important dispersion variables and to give an approximate result of the total dispersion, all with a reasonable amount of simulations.

The aim of this paper is to present a methodology that provides an analytical tool for estimation of robustness and response variation within a pre-defined process window. To exemplify the developed methodology, the stochastic simulation technique is used for a sheet-metal forming application. A sampling plan based on the Latin hypercube sampling method for variation of design parameters is utilized, and the thickness reduction is specified as the response. Moreover, the response surface methodology is applied for understanding the quantitative relationship between design parameters and response value. The conclusions of this study are that the applied method gives a possibility to illustrate and interpret the variation of the response versus a design parameter variation. Consequently, it gives significant insights into the usefulness of individual design parameters. It has been shown that the method enables us to estimate the admissible design parameter variations and to predict the actual safe margin for given process parameters. Furthermore, the dominating design parameters can be predicated using sensitivity analysis, and this in its turn clarifies how the reliability criteria are met. Finally, the developed software can be used as an additional module for set-up of stochastic finite element simulations and to collect the numerical results from different solvers within different applications.

One important issue when simulating tube hydroforming is to predict bursting, i.e. when the increase in cavity pressure cannot be compensated by hardening of the tube material. Traditionally, this is made by a forming limit diagram (FLD), where the limit strains determine whether a material point is experiencing necking or failure. However, the experimental FLD depends on the strain path, and the methods which are used to determine the FLD are adapted to conventional deep drawing which, depending on the process characteristics, could make it unsuitable for tube hydroforming applications. In this work, analytical and numerical forming limit predictions are studied from a hydroforming point of view. These predictions are then applied to free bulge cases, and a case with extensive feeding in a die where the results from the latter case is compared to experiments. Further, the influence from extrusion welds and a circumferential thickness distribution on the forming limit is also evaluated.

This paper describes the methodology used during the development of an energy absorbing Frontal Underrun Protection device (eaFUP). The aim of this study is to show how different optimisation methods can be used at different stages during the design process. It also shows one approach to derive an optimal design taking several different design alternatives into account, each of which consists of several different materials. The outcome of the optimisation process is three different designs of the eaFUP.

Tube hydroforming is gaining increasing interest from the metal forming industry. Complicated parts with a high level of structural component integration, e.g. engine cradles, subframes and exhaust systems, can be manufactured at a low cost with excellent repeatability. By using finite element (FE) simulations, there is a possibility to reduce the cost of expensive prototypes and reduce the trial and error design process to a minimum. However, when simulating a hydroforming process, the knowledge and computational methods used in conventional metal forming simulations are not always applicable. This concerns, e.g. the material modelling and validation. In this work, the influence of constitutive modelling on the results from a hydroforming process with extensive feeding is studied. In addition, interrupted tests have been used in order to validate the prediction of radial deformation and wall thickness throughout the complete process.

Tube hydroforming is a forming process where an inner pressure combined with axial feeding deforms the tube to the shape of a die cavity. One of the main concerns when designing such a process is to avoid burst pressure, i.e. the process state where the hardening of the material is unable to resist the increase in inner pressure and wall thickness reduction. The success of a hydroforming process strongly depends on the choice of process parameters, i.e. the combination of material feeding and inner pressure. Especially in hydroforming processes, where the free forming phase is substantial, the process is proved to be very sensitive to the inner pressure. By transforming the problem into a deformation controlled rather than a force controlled process, the results from the process parameter estimation become more reliable but on the other hand less intuitive. In this context, three distinct parameter estimation procedures are suggested. Firstly, a self feeding based procedure is proposed with the intention of being a fast method to be used as a first estimate of suitable process parameters. Secondly, an iterative optimization problem set up is presented. Thirdly, and finally, an adaptive simulation procedure based on process response approximations is proposed, which only requires a limited number of simulation runs.

When considering finite element simulations of aluminium tube hydroforming, the user is facing several challenges. Firstly, extruded aluminium is anisotropic in yield stress and plastic flow. Secondly, the hydroforming process introduces new issues concerning friction and process control. This imposes a demand for accurate constitutive models as well as for hydroforming process related testing methods. The present study focuses on how biaxial tests can be used to calibrate and validate a constitutive model. It is also shown that by using inverse modelling, additional information can be obtained through these types of tests, such as, e.g. the frictional behaviour for different lubrication conditions.

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.

Optimization of car structures is of great interest to the automotive industry. This work is concerned with structural optimization of a car body with the intent to increase the crashworthiness properties of the vehicle or decrease weight with the crashworthiness properties unaffected. In this work two different methodologies of constructing an intermediate approximation to the optimization problem are investigated, i.e. classical response surface methodology and Kriging. The major difference between the two methodologies is how the residuals between the true function value and the polynomial surface approximation value at a design point are treated.

Several different optimization problems have been investigated, both analytical problems as well as finite element impact problems.

The major conclusion is that even if the same kind of updating scheme is used both for Kriging and linear classic response surface methodology, Kriging improves the sequential behaviour of the optimization algorithm in the beginning of the optimization process. Problems may occur if a constraint is violated after several iterations and then classic response surface methodology seems to more easily be able to find a design point which satisfies the constraint.

In the present paper, an optimization technique has been used to minimize the risk of failure in a sheet metal forming process. Two different types of finite element solvers, one using total plasticity and the other using incremental plasticity, have been used. A comparison between response surface methodology and space mapping (SM) with the one-step solver as surrogate model has been done. The conclusion of this study is that the use of the total plasticity theory drastically reduces the required computing time. Furthermore, the solution from the SM optimization algorithm is close to the solution obtained by the incremental plasticity solver.

In the present paper optimization has been used to evaluate alternative sheet metal forming processes. Six process set-ups were first defined in a hierarchy of designs and optimization was then used to evaluate each forming process of these designs. The challenge in designing the forming process was to avoid failure in the material and at the same time reach an acceptable through thickness strain. The conclusions of this study is that there may exist a different process that can give an improved product for the desired geometry. This process might be impossible for the optimization algorithm to reach due to either a poor starting point or a not so wise process set-up.

Experiments and simulations of penetration and perforation of high performance concrete targets by steel projectiles have been carried out. A Doppler radar monitored the projectile pre-impact velocity history and a high-speed camera captured the projectile residual velocity. A continuum mechanical approach and the finite-element method were used for the simulations. The targets were modelled with the K&C concrete model. For perforation the computational results show acceptable agreement with the experimental results, but not for penetration. The article is concluded with suggestions on how to better model concrete material for the current application.

The increased interest in lightweight materials for automotive structures has also lead to a search for efficient forming methods that suit these materials. One attractive concept is to use hydroforming of aluminum tubes. The advantages of this forming method includes better tolerances, decreased number of parts and an increased range of forming options. By using FE simulations, the process can be optimized to reduce the risk for failure, i.e. bursting or wrinkling. However, extruded aluminum is highly anisotropic and it is crucial that the material model used for simulations is able to accurately describe this behavior. Also, tube hydroforming occurs predominantly in a biaxial stress state which should be considered in the material testing, where uniaxial tests are used extensively in the industry today. The present study accentuates the need for improved constitutive models. It is shown that a material model, which accurately describes the anisotropic behavior of aluminum tubes, can be obtained from simple and robust experiments.

The accuracy of different approximating response surfaces is investigated. In the classical response surface methodology (CRSM) the true response function is usually replaced with a low-order polynomial. In Kriging the true response function is replaced with a low-order polynomial and an error correcting function. In this paper the error part of the approximating response surface is obtained from “simple point Kriging” theory. The combined polynomial and error correcting function will be addressed as a Kriging surface approximation.

To be able to use Kriging the spatial correlation or covariance must be known. In this paper the error is assumed to have a normal distribution and the covariance to depend only on one parameter. The maximum-likelihood method is used to find the latter parameter. A weighted least-square procedure is used to determine the trend before simple point Kriging is used for the error function. In CRSM the surface approximation is determined through an ordinary least-square fit. In both cases the D-optimality criterion has been used to distribute the design points.

From this investigation we have found that a low-ordered polynomial assumption should be made with the Kriging approach. We have also concluded that Kriging better than CRSM resolves abrupt changes in the response, e.g. due to buckling, contact or plastic deformation.

In the present paper, an optimization of the draw-in of an automotive sheet metal part has been carried out using response surface methodology (RSM) and space mapping technique. The optimization adjusts the draw bead restraining force in the model such that the draw-in in the FE-model corresponds to the draw-in in the physical process. The conclusion of this study is that space mapping is a very effective and accurate method to use when calibrating the draw-in of a sheet metal process. In order to establish draw bead geometry from the draw bead restraining force a 2D-model was utilized. The draw bead geometry found showed good agreement with the physical draw bead geometry.

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.

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.