On the need for the use of error‐controlled finite element analyses in structural shape optimization processes

Abstract

This is the peer reviewed version of the following article: Ródenas, J. J., Bugeda, G., Albelda, J. and Oñate, E. (2011), On the need for the use of error-controlled finite element analyses in structural shape optimization processes. SUMMARYThis work analyzes the influence of the discretization error associated to the Finite Element (FE) analyses of each design configuration proposed by the structural shape optimization algorithms over the behavior of the algorithm. The paper clearly shows that if FE analyses are not accurate enough, the final solution provided by the optimization algorithm will neither be optimal nor satisfy the constraints. The need for the use of adaptive FE analysis techniques in shape optimum design will be shown. The paper proposes the combination of two strategies to reduce the computational cost related to the use of mesh adaptivity in evolutionary optimization algorithms: (a) the use of an algorithm for the mesh generation by projection of the discretization error, which reduces the computational cost associated to the adaptive FE analysis of each geometrical configuration and, (b) the successive increase of the required accuracy of the FE analyses in order to obtain a considerable reduction of the computational cost in the early stages of the optimization process.

Cartesian grid FEM (cgFEM): High performance h-adaptive FE analysis with efficient error control. Application to structural shape optimization

“…The discretization error of each individual must be controlled to ensure the convergence of the optimization process [1,2]. This justifies the use of adaptive procedures to ensure that the prescribed accuracy level is obtained with the minimum computational cost.…”

“…Estes dos característiques conferixen a la tecnologia cgFEM dels requisits necessaris per a augmentar l'eficiència del codi cgFEM respecte a codis comercials. Com s'indica en [1,2], per a garantir la convergència del procés d'optimització de forma estructural es necessita controlar l'error en cada geometria analitzada. En este sentit el codi cgFEM també incorpora els apropiats estimadors d'error.…”

“…The coordinates of the children elements, relative to the parent element coordinates, in the virtual numbering are shown in 2.2. Additionally the local element characterization, that is, the local element children numbering (1,2,3,4), the local sides numbering and the local nodes numbering are shown in Figure 2.2 for 2D and in Figures 2.3, 2.4 and 2.5 for 3D.…”

“…The use of these techniques can provide the required accuracy in the solution with optimized FE models where the number of elements has been minimized, thus reducing the computational cost of the analysis. In some cases, like in structural shape optimization problems, h-adaptive analysis is a must because inaccurate FE results can negatively affect the behavior of the optimization algorithm [2], leading to non-optimal solutions, reducing the convergence rate to the optimal solution or even preventing convergence.…”

“…As a result of this the user needs to check each mesh before running the analysis. Finally, in order to guarantee that the optimization process converges to a suitable geometry, we need to control the error in each analysis performed by the lower level [1,2,4], otherwise the "noise" introduced by the numerical errors will make the optimization process to converge to a non-optimal solution, decrease its convergence rate or even prevent convergence. This issue has not been adequately addressed in commercial codes preventing their use in optimization processes.…”