This presentation was made at the 2019 NAFEMS World Congress in Quebec Canada

Resource Abstract

An approach to the numerical simulation of a contact interaction between the deformable solids inside the assembly is considered in the article. Solids interact with each other during the deformation process without sliding and detachment along internal boundaries (in other words solids are merged to each other). A standard approach for solving such kind of problems is to imprint the boundaries and merge the solids along the common boundary zones. However this approach requires conformal discretization of the whole assembly including conformal meshes on the common boundary regions during the numerical simulation, which often causes significant problems for the industrial assemblies consisting of a large number of parts of different sizes. As a result a mesh could contain a large number of elements, also it is not possible to make a sharp transition from the coarse mesh to the detailed one, to connect meshes with different element types (tetrahedrons, hexahedrons), to generate an unstructured hexahedral mesh in the overall assembly. Moreover, it is difficult or impossible to build a conformal mesh in the overall assembly in case of imperfections (gaps, overlaps, etc. between solids) in the initial geometrical model (which often happens while importing CAD models into CAE systems), and if solids are not ideally attached to each other. As a result it is necessary to heal/modify an initial CAD-model (which is time consuming and not a straightforward process) to build the mesh of acceptable quality.



One of the approaches for solving the described problems with generating meshes for assemblies is to remove a mesh conformity requirement between solids and to build instead independent discretization in each solid with further tying in order to provide a continuous solution of the boundary-value elasticity problem (stress-strain state parameters) along the boundaries between the solids. A tying algorithm based on the bonded contact interaction between solids is described in the article. A bonded contact between the boundary elements inside the contact region is ensured by direct imposing displacement continuity conditions in the stiffness matrix (and a mass matrix, in case of transient problems) obtained from the discretization of a boundary-value elasticity problem inside the assembly. This is a direct generalization of an approach for setting Dirichlet conditions on displacements in the finite element method. Normal stress (traction) continuity in the contact regions is provided by the corresponding additional terms to the stiffness matrix from boundary integrals along the contact regions as a result of the Galerkin weak formulation (normal stresses continuity in a weak sense). High order space discretization is provided by the spectral element method. A described algorithm allows to obtain a numerical solution for the unstructured non conformal spectral element meshes (using different spectral element orders in solids), and to provide a continuity in C-norm for primary variables (displacements) and a continuity in L2-norm for normal stresses in the contact region.



Test examples are considered for the verification of the developed in CAE Fidesys (www.cae-fidesys.com) algorithm of tying elastic solids by comparing simulation results with the solutions of similar problems for the case of merged solids with a conformal mesh discretization: static and modal analysis of the assemblies consisting of cubic, cylindrical and spherical bodies. A robustness of the algorithm and a continuity of the obtained solution are analyzed in case of gaps/overlaps between contacting solids. It is shown that small gaps and overlaps in CAD-model of an assembly do not influence much a correctness of numerical results and these cases are correctly and automatically processed in CAE Fidesys software module based on the described algorithm. An example of an industrial problem of modal analysis of the micro sputnik composite part is considered.