BOUNDARY ELEMENT METHOD APPLICATION IN SOLIDS AND STRUCTURES

摘要

During the last 30 years or so, Boundary Element Methods (BEM) have been developed and successfully applied to the solution of a variety of engineering problems. BEM are at their best when applied to linear static or dynamic problems. However, during the last 15 years or so, BEM have also been successfully applied to the solution of a variety of materially and geometrically nonlinear problems. When non-linearities are present, the BEM loose their dimensionality reduction advantage over Finite Element Methods (FEM) since an interior discretization is necessary in addition to the boundary one. However, some advantages of BEM over FEM are still present.In this work, a boundary element method (BEM) formulation to perform linear bending analysis of building floor structures where slabs and beams can be defined with different materials is presented. The proposed formulation is based on Kirchhoff's hypothesis, the building floor being modeled by a zoned plate, where the beams are treated as thin sub-regions with larger rigidities. This composed structure is treated as a single body, the equilibrium and compatibility conditions being automatically taken into account. In the final integral equation, the tractions are eliminated along the interfaces, therefore reducing the number of degrees of freedom. The displacements are approximated along the beam cross-section, leading to a model where the values remain defined on the beam skeleton line instead of their boundaries. The accuracy of the proposed model is shown by comparing the numerical results with a well-known finite element code.The solid mechanics areas which have received special attention by researchers in the field during the last 5-10 years and which are considered in this course are the following:1) Special formulations and accurate and efficient numerical treatment of BEM, including symmetric formulations and computation of singular and hypersingular integrals. 2) Efficient treatment of materially and geometrically nonlinear static and dynamic problems, including inelastic (elastoplastic, viscoplastic,damage) behaviour, unilateral contact analysis and inelastic fracture mechanics. 3) Application to large static and dynamic problems of structural system analysis, including large three-dimensional systems, plates and shells and soil & end ash; structure interaction. 4) Application to various structural shape optimization and inverse or identification problems under both static and dynamic conditions, which are important in optimum structural design and nondestructive evaluation techniques. All the above categories of problems are presented in this course by specialists in the field.This method leads indeed to systems of equations with which the matrices are full, complex, nonsymmetrical, often badly conditioned and whose factorization is extremely expensive. In spite of this disadvantage, method BEM is often preferred with the finite element method because it is based on one formulation of the problem in the form of an integral of border, only radiant surface having to be discretized. This advantage of method BEM justified its use a long time almost exclusive in spite of is considerable data-processing cost.