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Variational method for construction of block-structured grids and thick prismatic mesh layers. , V.A. Garanzha 1,2 , L.N. Kudryavtseva 1,2 1 Computing Center RAS, Moscow 2 Moscow Institute of Physics and Technology. Contents.
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Variational method for construction of block-structured grids and thick prismatic mesh layers , V.A. Garanzha1,2, L.N. Kudryavtseva1,2 1Computing Center RAS, Moscow 2Moscow Institute of Physics and Technology
Contents • Variational mesh generation method based on hyperelasticity theory; • Mesh untangling technique; Applications: • Untangling and optimization of structured and block-structured meshes. • Construction of thick prismatic meshes using variational method;
Finite hyperelasticity methods for mesh generation • Elastic deformation is constructed as a mapping of domain in Lagrangian coordinates onto implicit domain in Eulerian coordinates. • Elastic deformation maps Cartesian net in Lagrangian coordinates onto curvilinear mesh; • Internal energy is minimized taking into account slip boundary conditions on implicitly defined boundary; • Finite element method is used to approximate hyperelastic energy; • Iterative energy minimization is based on preconditioned gradient search with projected gradients technique near boundaries.
Variational principle of hyperelasticity in lagrangian coordinates
Examples of polyconvex distortion measures Shape distortion measure. B.Joe, 1991, V. Liseikin, 1991, M. Rumpf, 1996, P.Knupp, 2001 Balanced distortion measure, θ = 4/5, Garanzha, 2000, Garanzha, Branets 2002, Branets, Carey 2003 Garanzha, 2000. Quasi-isometric distortion measure can be used for max-norm optimization of meshes and spatial mappings
Penalty formulation for barrier distortion measures 2d: Garanzha, Kaporin, 1999, 3d: Garanzha, Branets, 2002, Escobar et al, 2003 Earlier developments: S.Ivanenko, 1988, 1996, M. Rumpf, 1996, K. Rimslagh, 1996
Application of variational method for structured meshing: example of tesselated model
Successive untangling and optimization of 3d structured mesh
Resulting boundary orthogonal 3d mesh
Ansys mesh for similar but simpler configuration contains 63 blocks
Swept winged body – rather hard test for structured meshing and untangling
Construction of thick prismatic meshes using variational method ContentsObjectives of research Variational mesh generation method using hyperelasticity theory; Construction of thick prismatic layers using variational methods; Elimination of layer self overlap using rough approximation of medial surfaces; Variational method for refinement and orthogonalization of meshes in prismatic layer. Development of hybrid grid module for multiphysics software tool LOGOS; Efficient implementation for huge meshes; Development of automatic almost structured mesh generator.
Stages of prismatic mesh construction • Construction of relatively thin one-cell thick prismatic mesh near boundary; • Layer enlargement using elastic springback model – assuming that initial guess is highly compressed hyperelastic material with free boundary; • Elimination of layer self-overlaps by cutting excessive material; • Refinement and orthogonalization of 1-layered prismatic mesh using combination of variational method and marching technique.
Insensitivity of prismatic layer to mesh size and quality of surface elements
Construction of thick layer in the presence of thin passages and acute corners
Self-overlap resolved by constructing approximate medial surface
Outline of the algorithm, I: initial thin layer, successive springback stages, elimination of self-intersections
Outline of algorithm, II: precise thickness cut and smoothing, steps of variational marching technique, resulting prismatic layer
Initial surface and last surface of prismatic layer, depending on surface orientation
Example of the surface containing two non-lipschitz vertices
Prismatic layer in the neighborhood of non-lipschitz vertices inevitably contains degenerate prisms
Surface of the model contains more than a hundred nonlipschitz vertices
Prismatic layer in the presence of the nonlipschitz vertices
Prismatic layer thickness is comparable to the characteristic size of the model
Conclusions and directions of further research • Variational method is suggested with allows to construct thick prismatic layers around bodies of complicated shape; • Method can be applied when nonlipshitz vertices are present on the surfaces; • Efficiency issues for very large surface meshes should be resolved by applying variation method only in key regions; • We plan to integrate prismatic layer generator with tet-, adaptive cartesian and polyhedral meshing tools; • We plan to use this technique as building block for almost structured automatic mesh generator
Twisted prism as a result of numerical springback simulation