Bruno Barthelemy

Accuracy problems associated with semi-analytical derivatives of static response

Abstract The semi-analytical method is widely used for calculating derivatives of static response with respect to design variables for structures modeled by finite elements. This paper shows that the method can have serious accuracy problems for shape design variables in structures modeled by beam elements. The problem was first discovered in the analysis of a complex car model which is described. Next, it is shown that same accuracy problem occurs for the simplest beam structures. Finally, the problem is shown to be associated with a high ratio of rigid body rotation to elastic deformation, and a test is developed to diagnose the severity of the problem in a given structure. Several examples are presented to show the validity of this test.

On the accuracy of shape sensitivity

The calculation of sensitivity of the response of a structure modeled by finite elements to shape variation is known to be subject to numerical difficulties. The accuracy of a given method is typically measured against the yard stick of finite-difference sensitivity calculation. The present paper demonstrates with a simple example that this approach may be flawed because of discretization errors associated with the finite element mesh. Seven methods for calculating sensitivity derivatives are compared for a two-material beam problem with a moving interface. It is found that as the mesh is refined, displacement sensitivity derivatives converge more slowly than the displacements. Six of the methods agree fairly well, but the adjoint variational surface method provides substantially different results. However, the difference is found to reflect convergence from another direction to the same answer rather than reduced accuracy. Additionally, it is observed that small derivatives are particularly prone to accuracy problems.

Physically based sensitivity derivatives for structural analysis programs

Most formulations for sensitivity derivatives of structural response require detail-level computations (e.g. element-level calculations for finite-element systems). This requirement is a major difficulty in this age of “black box” commercial structural-analysis software which typically does not provide the user with access to that level of detail. A recent formulation of sensitivity analysis by Mróz and Dems provides a solution to this difficulty. The sensitivity calculations only require the solution of the original problem with new imposed loading consisting of initial stresses and initial strains. Therefore, they require only that the structural analysis program allow these type of loading. The present paper demonstrates the application of this approach to one and two-dimensional problems, using a finite-element program and a nonlinear shell analysis program.

On the accuracy of shape sensitivity derivatives

Six methods for calculating shape sensitivity derivatives are compared for a two-material beam problem with a moving interface. It is found that as the finite-element mesh is refined, displacement sensitivity derivatives converge more slowly that the displacement themselves. Five of the methods agree fairly well, but the adjoint variational surface method provides substantially different results. However, the difference is found to reflect convergence from another direction to the same answer, rather than reduced accuracy.

Self-Pierced Rivet (SPR) Modeling in Aluminum Structure Crash Analysis

A finite element method has been developed for modeling the characteristics of self-pierced rivets (SPR) in the aluminum parts and structures subjected to impact. It considers SPR elastic, inelastic and separation behaviors and was developed based on the coupon test results in lap shear, coach peel, and tension modes. The method consists of establishing baseline strength characteristics in the six degrees of freedom; and then modifying the baseline properties with coefficient factors (Ki) factors that influence the performance of SPR (thickness, differences in material, temperature, impact velocity, size of connection). The resulting combination is a material damage function that is utilized as the material properties of beam-type spring elements simulating the SPR connection in a FE model. The baseline properties and the equations for coefficient factors (Ki) developed in this study are provided. The material damage functions were developed in the RADIOSS environment, but can be applied to other codes as well. The method was validated at the component level using various configuration of a hat section tube. The CAE prediction using the new method correlates with the test results well in all the cases.© 2002 ASME

Integrated structural analysis and design using 3-D finite elements

The optimum design of the shape of a hole in a plate in tension is obtained with a design algorithm based on partially converged analysis. The plate is modeled by three-dimensional finite elements, and an EBE PCG iterative solution algorithm is used for solving the equations of equilibrium. Several parametrizations of the optimum shape are considered, and it is found that a sine series can describe the optimum shape with only three design variables. The optimum shape compares well with an experimental optimum obtained by Durelli.

Physically based sensitivity derivatives for finite-element programs

Attention is given to a virtual-work approach to the definition of sensitivity in terms of physical quantities; sensitivity fields are treated as a solution to loadings that consist of body forces, initial strains, and initial stresses. The method can accordingly be implemented in a finite element package, with only minimal knowledge of its programming detail and without access to its source code. The application of the approach to one- and two-dimensional problems is demonstrated. The sensitivity of the structural response for truss structures was computed with respect to the cross-sectional areas of the members.