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

Properties of hyperelastic potential

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

Global flattening and curvature sensitive planar remeshing.

Result of remeshing is mapped back to model surface

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

Coordinate surfaces for winged body test case

Construction of thick prismatic meshes using variational method

Contents Objectives 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.

Stages of prismatic mesh construction

Insensitivity of prismatic layer to mesh size and quality of surface elements

Prismatic layer behaviour inside acute corners

Construction of thick layer in the presence of thin passages and acute corners

Layer self-overlap zone

Self-overlap resolved by constructing approximate medial surface

Realistic aircraft model

Layer self-overlap zone

Approximate medial surface is constructed

Resulting self-contact spot is shown in yellow

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

Outer boundary of prismatic layer

Rafal test case

Initial surface and last surface of prismatic layer, depending on surface

orientation

Interior and exterior prismatic layers

Interior and exterior layer around camel mouth

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

Project of CAGI spacecraft

Example of very thick layer

Prismatic layer thickness is comparable to the characteristic size of the model

Test hybrid mesh around spacecraft

Intermediate stage of semi-structured mesh construction

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

Twisted prism: isolated view