1. 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

2. 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;

3. 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.

4. Variational principle of hyperelasticity in lagrangian coordinates

5. Properties of hyperelastic potential

6. 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

7. 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

8. Application of variational method for structured meshing: example of tesselated model

9. Global flattening and curvature sensitive planar remeshing.

10. Result of remeshing is mapped back to model surface

11. Successive untangling and optimization of 3d structured mesh

12. Resulting boundary orthogonal 3d mesh

13. Ansys mesh for similar but simpler configuration contains 63 blocks

14. Swept winged body – rather hard test for structured meshing and untangling

15. Coordinate surfaces for winged body test case

16. 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.

17. 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.

18. Stages of prismatic mesh construction

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

20. Prismatic layer behaviour inside acute corners

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

22. Layer self-overlap zone

23. Self-overlap resolved by constructing approximate medial surface

24. Realistic aircraft model

25. Layer self-overlap zone

26. Approximate medial surface is constructed

27. Resulting self-contact spot is shown in yellow

28. Outline of the algorithm, I: initial thin layer, successive springback stages, elimination of self-intersections

29. Outline of algorithm, II: precise thickness cut and smoothing, steps of variational marching technique, resulting prismatic layer

30. Outer boundary of prismatic layer

31. Rafal test case

35. Initial surface and last surface of prismatic layer, depending on surface orientation

36. Interior and exterior prismatic layers

37. Interior and exterior layer around camel mouth

38. Example of the surface containing two non-lipschitz vertices

39. Prismatic layer in the neighborhood of non-lipschitz vertices inevitably contains degenerate prisms

40. Surface of the model contains more than a hundred nonlipschitz vertices

41. Prismatic layer in the presence of the nonlipschitz vertices

42. Project of CAGI spacecraft

43. Example of very thick layer

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

45. Test hybrid mesh around spacecraft

46. Intermediate stage of semi- structured mesh construction

47. 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

48. Twisted prism as a result of numerical springback simulation

49. Twisted prism: isolated view