Pierre Beckers

Dual analysis with general boundary conditions

Abstract The dual analysis concept was introduced by Fraeijs de Veubeke [1] as a consequence of upper and lower bounds of the energy. As such bounds exist only in those cases where one type of condition is homogeneous, it is commonly admitted that a dual error measure does not exist with general boundary conditions. This paper presents a re-examination of the dual error measure by a way which avoids any use of upper and lower bounds of the energy. It is found that such an error measure holds whatever the boundary conditions be. Furthermore, it is not necessary to obtain the approximate solutions by a Rayleigh-Ritz process, so that the second analysis, which seemed necessary in the original dual analysis concept, may be replaced by any admissible approximation. This implies the possibility of a dual error measure at a simple post-processor level.

Structural design of aircraft skin stretch-forming die using topology optimization

This paper demonstrates a topology optimization method for the large scale stretch-forming die design. The implementation of the topology optimization depends on the surface loads obtained from the numerical simulation of the sheet metal stretch-forming process. Typically, the design variables are defined based on a linear interpolation to generate directional structures satisfying the manufacturing requirement. To further validate the design procedure, mathematical derivations have proved that the design sensitivities are strictly continuous. More explanations and numerical tests are presented to show the variation of the surface loads versus the stiffness of the die. A stretch-forming die design example is solved on account of gravity and surface loads. The final solution is compared with the traditional design. The numerical results have shown that the topology design can improve the stiffness and strength of the stretch-forming die significantly.

On the multi-component layout design with inertial force

The purpose of this paper is to introduce inertial forces into the proposed integrated layout optimization method designing the multi-component systems. Considering a complex packing system for which several components will be placed in a container of specific shape, the aim of the design procedure is to find the optimal location and orientation of each component, as well as the configuration of the structure that supports and interconnects the components. On the one hand, the Finite-circle Method (FCM) is used to avoid the components overlaps, and also overlaps between components and the design domain boundaries. One the other hand, the optimal material layout of the supporting structure in the design domain is designed by topology optimization. A consistent material interpolation scheme between element stiffness and inertial load is presented to avoid the singularity of localized deformation due to the presence of design dependent inertial loading when the element stiffness and the involved inertial load are weakened with the element material removal. The tested numerical example shows the proposed methods extend the actual concept of topology optimization and are efficient to generate reasonable design patterns.

Numerical comparison of several a posteriori error estimators for 2D stress analysis

ABSTRACT The reliabilities of several a posteriori error estimators, including those of Gago, Zienkiewicz-Zhu, and more recently those proposed by Beckers and Zhong are compared through a set of examples in plane elasticity. The examples range from those having analytic solutions to those having progressively stronger singularities. The examples generally use either 4 (or 8) node quadrilaterals for initial comparisons. The results of these examples indicate that, in certain cases, some of the error estimators are unreliable and do not appear to be asymptotically exact. Further studies are suggested to investigate the general validity of the initial conclusions.

A general rule for disk and hemisphere partition into equal-area cells

A new general rule is presented to define procedures in order to cut a disk or a hemisphere into an imposed number of equal-area cells. The system has several degrees of freedom that can be fixed, for instance, by enforcing the cells aspect ratios. Therefore, the cells can have very comparable forms, i.e. close to the square. This kind of method is effectively useful because it is not possible to build exact dense uniform distributions of points on the sphere. However, it will be shown that it is easy to cover the sphere, the hemisphere or the disk with equal-area cells. This capability makes easy, for instance, the implementation of stratified sampling in Monte Carlo methods. Moreover, the use of different azimuthal projections allows to link problems initially stated either on the hemisphere or within the circle.

RECENT DEVELOPMENTS IN SHAPE SENSITIVITY ANALYSIS: THE PHYSICAL APPROACH

Because of increasing CAD requirements in industry, shape optimum design by finite element is an important field of research and development. The global solution of this problem involves two main aspects: the first one is the availability of powerful and robust optimizers and the second one is the necessity of producing the best sensitivity analysis for the approximate problem concerned by optimization. Nowadays, several good softwares or computer programs are available in order to solve a well stated approximate optimization problem. For sensitivity analysis the method using big design elements is quite successful but it appears that more general and powerful methods could be implemented. It is expected that the physical approach developed in this paper could be one of these better methods of the future.

A posteriori error estimators related to equilibrium defaults of finite element solutions for elastostatic problems

Abstract This paper presents a study of two types of a posteriori error estimator based on equilibrium defaults in finite element solutions of planar linear elastic problems. The G -type estimator, developed at LTAS-Inforgraphie, uses explicit relations between the equilibrium defaults and the energy of the error. This type is represented here with a new physical interpretation, and with the reference tractions on the sides of elements redefined so as to satisfy equilibrium. The δ-type estimator uses a reference solution for stress δ, which has no equilibrium defaults, i.e. δ is a statically admissible stress field. The reference solutions considered in this paper are based on the formulation first proposed by Ladeveze. This present study considers several alternative formulations of equilibrating tractions suitable for both types of estimator, and its is restricted to errors in models composed of 4-noded bilinear Lagrange elements. Numerical results are presented for several examples with smooth or singular stress fields. It is observed that significant improvements to the effectiveness of the G -estimator can be obtained, particularly for coarser meshes. With judicious choice of equilibrating tractions it is concluded that this estimator can be more effective than the δ-estimator.

The universal projection for computing data carried on the hemisphere

For many years, important efforts have been devoted to efficiently compute the view factors in the frame of radiative heat exchanges. This subject has also received special attention in the more recent development of global illumination methods. Basically two techniques are available; the first one is based on projections and the second one on ray tracing methods. Here we will present a new algorithm dealing with projections and show that we can solve both the problems of computing view factors and solid angles by using the same projection. The proposed method is precise and fast, so it can be used in interactive design software.

A unified approach for displacement, equilibrium and hybrid finite element models in elasto-plasticity

Abstract Finite element models for elasto-plastic incremental analysis are derived from a three-field variational principle. The Newton-Raphson method is applied to solve the nonlinear system of equations which is obtained from the stationarity condition of this principle. The iterative schemes are discussed in detail for pure displacement and for pure equilibrium models from which iterative schemes for hybrid models follow directly. In the displacement model, the compatibility of the strains and the plasticity criterium are satisfied during the whole iterative process, while the equilibrium of the stresses is restored only in the mean after convergence. In the equilibrium model, the plasticity criterium and the compatibility of the strains are verified in the mean during the iterative process; when convergence is achieved, the stresses are locally in equilibrium with the applied external loads. In both cases, a tangential stiffness matrix can be constructed, even for perfectly plastic materials and it allows one to obtain always very good convergence properties. Examples are shown for plane stress and axisymmetric cases.

Use of geometry in finite element thermal radiation combined with ray tracing

In heat transfer for space applications, the exchanges of energy by radiation play a significant role. In this paper, we present a method which combines the geometrical definition of the model with a finite element mesh. The geometrical representation is advantageous for the radiative component of the thermal problem while the finite element mesh is more adapted to the conductive part. Our method naturally combines these two representations of the model. The geometrical primitives are decomposed into cells. The finite element mesh is then projected onto these cells. This results in a ray tracing acceleration technique. Moreover, the ray tracing can be performed on the exact geometry, which is necessary if specular reflectors are present in the model. We explain how the geometrical method can be used with a finite element formulation in order to solve thermal situation including conduction and radiation. We illustrate the method with the model of a satellite.