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# lecnotes23 - 13.472J/1.128J/2.158J/16.940J COMPUTATIONAL...

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13.472J/1.128J/2.158J/16.940J COMPUTATIONAL GEOMETRY Lecture 23 Dr. W. Cho Prof. N. M. Patrikalakis Copyright c 2003 Massachusetts Institute of Technology Contents 23 F.E. and B.E. Meshing Algorithms 2 23.1 General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 23.1.1 Finite Element Method (FEM) . . . . . . . . . . . . . . . . . . . . . . 2 23.1.2 Mesh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 23.1.3 Some Criteria for a Good Meshing . . . . . . . . . . . . . . . . . . . . 3 23.1.4 Finite Element Analysis in a CAD Environment . . . . . . . . . . . . . 4 23.1.5 Background Information . . . . . . . . . . . . . . . . . . . . . . . . . . 4 23.1.6 Mesh Conformity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 23.1.7 Mesh Refinement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 23.2 Mesh Generation Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 23.2.1 Mapped Mesh Generation [23, 10, 11, 5] . . . . . . . . . . . . . . . . . 10 23.2.2 Topology Decomposition Approach [20, 3] . . . . . . . . . . . . . . . . 12 23.2.3 Geometry Decomposition Approach [6, 17, 13] . . . . . . . . . . . . . . 13 23.2.4 Volume Triangulation – Delaunay Based [7, 15] . . . . . . . . . . . . . 14 23.2.5 Spatial Enumeration Methods [21, 22] . . . . . . . . . . . . . . . . . . 15 Bibliography 16 1

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Lecture 23 F.E. and B.E. Meshing Algorithms 23.1 General 23.1.1 Finite Element Method (FEM) FEM solves numerically complex continuous systems for which it is not possible to construct any analytic solution, see Figure 23.1. Physical Problem Law of Physics PDE Weighted Residual Methods Integral Formulation Algebraic System of Equations Unknown Function Approximation (using finite element mesh and matrix organization) Approximate Solution Numerical Solution Formulation of Equations Transformation of Equations Solution Figure 23.1: FEM Procedure 2
23.1.2 Mesh Mesh is the complex of elements discretizing the simulation domain, eg. triangular or quadri- lateral mesh in 2D, tetrahedral or hexahedral mesh in 3D. Why Use Mesh ? To construct a discrete version of the original PDE problems. Types of Mesh See Figure 23.2. Figure 23.2: Structured ( n = 6) and unstructured ( n 6 = constant ) mesh Structured mesh The number of elements n surrounding an internal node is constant . The connectivity of the grid can be calculated rather than explicitly stored . simpler and less computer memory intensive Lack of geometric flexibility Unstructured mesh The number of elements surrounding an internal node can be arbitrary . Greater geometric flexibility crucial when dealing with domains of complex geometry or when the mesh has to be adapted to complicated features of computational domain or to complicated features of the solution (eg. boundary layers, internal layers, shocks, etc). Expensive in time and memory requirements 23.1.3 Some Criteria for a Good Meshing Shape of elements [2, 18] meshing should avoid both very sharp and flat angles, see Figure 23.3. may cause serious numerical problems in both finite element mesh generation and analysis. 3

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Figure 23.3: Elements with sharp and flat angles Number of elements should be moderate. related to efficiency of finite element analysis. Topological consistency (homeomorphism) between the exact input domain and its mesh . related to robustness of finite element analysis [8, 9]. Automation, adaptability, etc. 23.1.4 Finite Element Analysis in a CAD Environment See Figure 23.4.
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