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A Formulation of Generalized Node , a Quadratic Tetrahedron/Triangle with only Corners, and Meshfree Approximation with

R. Tian G. Yagawa Center of promotion for Computational Science and Engineering (CCSE), JAERI. The 8th US National Congress on Computational Mechanics(USNCCM8) Austin, TX, July 25-27 2005.

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A Formulation of Generalized Node , a Quadratic Tetrahedron/Triangle with only Corners, and Meshfree Approximation with

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  1. R. Tian G. Yagawa Center of promotion for Computational Science and Engineering (CCSE), JAERI The 8th US National Congress on Computational Mechanics(USNCCM8) Austin, TX, July 25-27 2005 A Formulation of Generalized Node, a Quadratic Tetrahedron/Triangle with only Corners, and Meshfree Approximation with Rotational DOFs R.Tian

  2. A brief review of recent numerical methodologies • Meshless methods: no mesh and meshing; simple model data; mesh distortions • Particle methods: large geometrical changes; mesh distortion • Generalized FEM: • Can be meshless  • Can be FEM  • PUM, FEPUM (Babuska and his cooperators 1997): general PU framework • hp cloud, cloud based FEM (Oden and his cooperators. 1996, 1998): hp adaptivity • X-FEM (Belytshko and his cooperators. 1997): discontinuity representation in function space • GFEM (Strouboulis et al. 2000, Duarte et al. 2000): handbook functions; 2D=>3D • FEM => GFEM, two issues of GFEM • Avoiding linear dependencies (Tian, Yagawa, Terasaka. CMAME, 2005) • Recovering physical meaning of nodal unknowns (Tian, Yagawa. IJNME, 2005) R.Tian

  3. What is Generalized Node? • Based on Generalized Finite Element Methods (GFEM): • PUM (Babuska, Melenk, 1997); cloud based FEM (Oden et al. 1998); GFEM (Strouboulis et al. 2000, Duarte et al. 2000); Manifold Method (Shi 1991) R.Tian

  4. What is GFEM and what is the main merit? Elems: 512 Nodes: 297 Elems: 2 Nodes: 4 Num/Exa: 94.14% Num/Exa: 99.69% • Achieving good accuracy using cheap mesh; • Remarkably reducing mesh generation and computing effort. R.Tian

  5. What is Generalized Node? • Generalized Finite Element Methods (GFEM): • PUM (Babuska, Melenk, 1997); cloud based FEM (Oden et al. 1998); GFEM (Strouboulis et al. 2000, Duarte et al. 2000); Manifold Method (Shi 1991) • Main merit: • Increasing approximation order without necessarily adding new nodes tothe underlying mesh. • Generalized node is a physically defined local functionf. R.Tian

  6. Why use Generalized Node? • Issues in Generalized Finite Element Methods: • Linear dependence problem (singular stiffness matrix); • Physically meaningless extra nodal dofs. • We are focused on: • Avoidable linear dependencies • Triangular and tetrahedral meshes are always superior to other mesh types in designing a stable GFEM*; • Physically meaningful nodal unknowns • Possible connections with existed elements; • Boundary treatment. *Tian, Yagawa, Terasaka. Computer Methods in Applied Mechanics and Engineering 2005. R.Tian

  7. Why use Generalized Node?3D complicated finite element structure analysis • Practical 3D structures or mechanical components: • Complex geometries; • Construction block of arbitrary geometries: tetrahedrons. • 3D large meshes: • Fully automatic tetrahedral mesh generators. • Tetrahedrons: most practical element type for complex 3D finite element structure analysis. R.Tian

  8. Why use Generalized Node?easy mesh generation vs. good accuracy Element Mesh generation Accuracy Bad Good Fully automatic mesh generators TET4 Good accuracy tetrahedron element is favorable! goodExcellent general purpose element Well but badin re-meshing TET10 goodExcellent general purpose element Bad HEX20 TET4: Tetrahedron HEX: Hexahedron R.Tian

  9. Why use Generalized Node?Current advanced tetrahedrons • TET4RX: Pawlak et al. 1991.IJNME1991; 31:593-610. HT4R18, HT4R14: Sze and Pan 2000. IJNME. 2000; 48(7):1055-1070. • Using Allman’s rotational DOFs. • Containing stabilization parameters. • Intermediate accuracy lying between TET4 and TET10. IJNME: Int.J.Numer.Methods Engrg. DOF: Degree Of Freedom R.Tian

  10. Based on Manifold Method (Shi, 1991) and PUM (Babuska et al. 1997) PUM: Partition of Unity Method 1. Generalized Node R.Tian

  11. Four components of infinitesimal deformation 1.2 Generalized Node (GN) Conventional nodes (1) Rigid body translations. (2) Rotations around node. (3) Simple tension/compression. (4) Pure shearing. A generalized node New nodal d.o.fs Nodal displacements R.Tian

  12. A full order GN An intermediate order GN A zeroth order GN 1.3Elementswith generalized node-- Hierarchical structureof generalized node Mesh Generalized nodes Relations Full quadratic Equivalentto elements with Allman’s rotation Linear elements -- special cases Same mesh but different order!! R.Tian

  13. 2. GN based approximation: MM(Shi, 1990) PUM(Babuska, et al, 1996) u, uh u, uh x x Approximation method Mesh based Meshfree Interpolation Approximation Part 1: High Accuracy Elements Part 2: High Accuracy MLS withrotational DOFs R.Tian

  14. GN based approximation: applications • Part I: high accuracy elements with only corners • A 3-node quadratic triangle: GNT3 • A 4-node quadratic tetrahedron: GNTet4 • Part II: high accuracy moving least square approximation R.Tian

  15. Part 1. High accuracy elements: Selected Numerical Examples • Labels • GNT3: New 3-node triangle • GNTet4: New 4-node tetrahedron • Benchmarks • (1) Eigenvalue analyses –stability assessment • Constant strain patch test: consistency • (2) High order patch test: order assessment • Volume locking test: robustness test • Shearing locking test: robustness test • Hole in an infinite hole: convergence studies • (3) Cook’s problem: comparisons with advanced triangles • (4) Curved cantilever beam problem: accuracy examination • MacNeal and Harder slender beam problem • (5) Twisted beam problem: accuracy examination • (6) Sensitivity to mesh distortion • (7) Computational effort estimation R.Tian

  16. (1) Eigenvalue analyses-- No spurious zero-energy modes Background: ♣ Plane element: 3 rigid body modes => 3zero zero-eigenvalue of stiff. Matrix. ♣ Essential (displacement) boundary => eliminate all rigid body modes => stiff Matrix is nonsingular ♣ Spurious zero-energy modes => stiff. Matrix is singular => unstable element No spurious zero-energy modes. Linear dependencies eliminated! R.Tian

  17. (2) High order patch test: GNT3 – complete quadratic order (1) Order assessment ►Solution of a cantilever under bending load = quadratic ►Reproducing exact solution of this problem => an element at least is of order 2. (2) High order patch test (GNT3) (3) Results of high order patch tests GNT3: a complete quadratic R.Tian

  18. (3) Cook’s problem: Compare GNT3 with existing triangular elements R.Tian

  19. Normalized solutions of Cook’s problem. GNT3 ≈ T6, Q8 > others R.Tian

  20. (4) Curved beam problem: GNTet4 R.Tian

  21. (4)Normalized solutions for curved beam. TET10 HEX20 GNTet4 > HEX20 > TET10 >> TET4 R.Tian

  22. (5) Twisted beam: GNTet4 R.Tian

  23. TET10 0.987 0.996 (5) Normalized solutions for twisted beam Element In-plane loading Out-of-plane loading TET4 0.096 0.080 TET4R 0.428 0.381 TET4RX 0.583 0.477 HT4R18 0.617 0.623 HT4R14 0.759 0.729 GNTet4 0.999 1.008 HEX8 0.206 0.332 HEX8X 0.992 0.986 HEX8R 0.951 0.960 HEX8RX 1.001 0.999 HEX20 0.944 0.957 Exact value 1.0 1.0 GNTet4 > TET10 > HEX20 >> TET4 R.Tian

  24. (6) Mesh distortion: GNTet4 R.Tian

  25. (6) Normalized solutions of cantilever with distorted mesh Sensitivity to mesh distortion: GNTet4 >TET10 >> HEX20 R.Tian

  26. GNTet4: new 4-node tetrahedron HEX20:20-node hexahedron, quadratic, midside nodes TET10:10-node tetrahedron, quadratic, midside nodes TET4 : 4-node tetrahedron, linear hbw : half band width dof: degree of freedom (7) GNTet4, HEX20, TET10, TET4 (1) (3) Accuracy vs. computational efforts GNTet4: dofs:108 hbw:72 Tet4: dofs:3075 hbw: 81 (2) Accuracy vs. mesh generation GNTet4: nodes:12 elems:12 Tet4: nodes:1025 elems:3840 To achieve given accuracy: GNTet4 > HEX20 > TET10 >> TET4 (mesh generation) GNTet4 ≈ HEX20 > TET10 >> TET4(computing-wise) R.Tian

  27. GN based approximation: application • Part I: high accuracy elements with only corners • A 3-node quadratic triangle: GNT3 • A 4-node quadratic tetrahedron: GNTet4 • Part II: high accuracy moving least square approximation • Moving least squares approximation • PDE solving using nodal data only. • Surface modeling; • Scatted data modeling, fitting etc; R.Tian

  28. Current MLS approximations Linear basis Quadratic basis (1 )Small bandwidth (2) Matrix A: 3×3 Computationally cheap Mostly used (1) Large bandwidth (2) Matrix A: 6×6 Computationally expensive Seldom used make essential boundary too stiff when penalty method used a GN-based MLS achieving accuracy of quadratic basis R.Tian

  29. Part 2:GN-basedMLS approximation dofs/node 2D 3D Comments New MLS approximation Even better than quadratic basis 6 12 A MLS approximation with nodal rotations [Tian, Yagawa,2004] 3 6 Standard MLS approximation -- a special case 2 3 : constructed using the linear basis, always cheap to compute!! R.Tian

  30. An Example: A high accuracy MLS approximation using only linear basis. Beam Benchmark Generalized node: effective for meshfree approximation Conventional nodes for easy boundary treatments Accuracy vs. nodes High order generalized nodes Accuracy vs. dofs R.Tian

  31. Conclusions and remarks • A formulation of Generalized Node (GN) was presented and employed to develop: • A new 3-node quadratic triangle; • A new 4-nodequadratic tetrahedron; • A new MLS approximation with rotational dofs. • One of major features of the generalized node is its hierarchical structure. (to be continued) For more details: R. Tian, G. Yagawa. Generalized nodes and high performance elements. International Journal for Numerical Methods in Engineering. To appear R.Tian

  32. Hierarchical structure of generalized node Mesh Generalized nodes Relations GNT3 / GNTet4 A full order GN Equivalentto elements with Allman’s rotation An intermediate order GN Linear elements -- special cases A zeroth order GN Same mesh, different order! R.Tian

  33. GNTet4: a new 4-node quadratic tetrahedron Element Mesh generation Accuracy Good Good GNTet4 Good Excellent general purpose element Well But badin re-meshing TET10 GoodExcellent general purpose element Bad HEX20 R.Tian

  34. R.Tian

  35. Hierarchical structure of generalized node Linear elements -- special cases A zeroth order GN An intermediate order GN Elements with Allman’s rotation A full order GN GNT3 / GNTet4 Same mesh but different order!! R.Tian

  36. Remarks • A concept of Generalized Node (GN) was presented. • The following high accuracy mesh-based ormeshfree approximation were constructed using the GNs, For more details: R. Tian, G. Yagawa. Generalized nodes and high performance elements. International Journal for Numerical Methods in Engineering. To appear R.Tian

  37. Relations with existing elements New triangles/tetrahedrons GNs Relations Equivalent to the elements with Allman’s rotation ≡Allman’s triangle, TET4RX& HT4R Conventional linear elements -- a special case R.Tian

  38. Estimation of computational efforts: GNTet4 vs TET10 and HEX20 Dofs vs mesh (computational efforts) Nodes vs mesh (mesh generation cost) For a very large mesh n: Tet10: maximum 8 times nodes of GNTet4 maximum 2 times dofs of GNTet4 Hex20:maximum 7 times nodes of GNTet4 maximum 1.5 times dofs of GNTet4 GNTet4: Easier and cheaper in both mesh generation and computing R.Tian

  39. Hole in an infinite plate: GNT3 shows same performance as T6 and Q8. GNT3 (18 dofs, 3 nodes) T6 (12 dofs, 6 nodes) Q8 (16 dofs, 8 nodes) Accuracy comparison Convergence rate in energy R.Tian

  40. Zero-displacement modes, essential boundaries and linear dependencies Extra zero energy modes could be derived by which also induce zero displacements, “zero-displacement modes”. These zero displacement modes can be suppressed by setting extra dofs to constant value, say zero, NO spurious zero energy modes arise. R.Tian

  41. Existing elements ×: bad, ○: good, ◎: excellent Element Mesh generation Accuracy × Generally NOT recommended unless very fine mesh used Intermediate accuracy TET4<TET4RX/HT4R<<TET10 TET4 ◎ Fully automatic mesh generator TET4 with Allman’s rotation TET4RX: Pawlak et al 1991 HT4R: Sze et al 2003 ◎Excellent general purpose element ○ × in remeshing TET10 ◎Excellent general purpose element × HEX20 Although Rome was built from blocks, in CAE world, this is unfortunately not true. R.Tian

  42. u, uh u, uh x x A simple outline of Mesh-based or meshfree methods PDEs Weighted Residual Galerkin Method Approximation method Approximation Lagrange Interpolation Mesh-based methods Meshfree methods R.Tian

  43. P (t ) P ( t ) P0=100 0 t Dynamic analysis • Wave propagation Click here Stress time history at the midpoint of the beam Using lumped mass matrix, the new element = the linear element R.Tian

  44. A, E, r L • Structural dynamic analysis – modal analysis Convergence of the fundamental frequency Using consistent mass matrix, the new element = the high order element with mid-side nodes. R.Tian

  45. Introduction (3)Current advanced tetrahedrons • TET4RX: Pawlak et al. 1991.IJNME1991; 31:593-610. HT4R18, HT4R14: Sze and Pan 2000. IJNME. 2000; 48(7):1055-1070. • Using Allman’s rotational DOFs. • Containing stabilization parameters. • Accuracy lying between TET4 and TET10. • Generalized FEM: Duarte et al. 2000.Comp.& Struct. 2000; 77(2): 215-232. IJNME 2002; 55: 1477-1492. • Using the notion of partition of unity (Babuska and Melenk 1997). • Linear dependence problem => singular stiffness matrix. • Requiring special equation solving method. • Physical meaning of extra nodal DOFs. IJNME: Int.J.Numer.Methods Engrg. DOF: Degree Of Freedom R.Tian

  46. Based on Manifold Method (Shi, 1991) and PUM (Babuska et al. 1997) PUM: Partition of Unity Method 1.High accuracy FEs  Large scale 3D FEA: Mesh generation  sequential vs. parallel solvers = bottleneck Free Mesh Method: a parallel mesh generator > Only linear triangle/tetrahedron Tetrahedron: de-facto element in 3D > Linear = bad accuracy, locking > Mid-side nodes = Re-meshing issue = Difficulty in Free Mesh Method High accuracy finite elements with only corner nodes R.Tian

  47. 1.1 Generalized node and high order elements Current advanced tetrahedrons = TET4RX: Pawlak et al. 1991; = HT4R: Sze et al. 2000 => Using Allman’s rotational dofs. Current Allman’s rotational dofs = Intermediate accuracy. TET4RX& HT4R (1) A quadratic 4-node tetrahedron Special cases (3) A much simpler but equivalent formulation of the element with Allman’s rotations (2) Linear counterpart Tian, Yagawa, Int. J. Numer. Methods Engrg., 2005 R.Tian

  48. Conventional node 1.3 Relationships with current tetrahedrons (1) FEM: Manifold Method(Shi,1991), PUM (Babuska et al. 1997) (2) Generalized node = Physically defined polynomial local approximations Linear dependence problem Singular stiffness matrices MM (Shi, 1991) PUM (Babuska et al. 1997) Generalized node Node with Allman’s rotations Conventional Tet4 Equivalent to TET4R (Pawlak 1991), but no spurious zero-energy mode. The new quadratic 4-node tetrahedron (or the new 3-node triangle) R.Tian

  49. 1.2 Generalized Node (GN) Four components of infinitesimal deformation Conventional nodes (1) Rigid body translations. (2) Rotations around node. (3) Simple tension/compression. (4) Pure shearing. A generalized node New nodal d.o.fs Nodal displacements R.Tian

  50. Four components of infinitesimal deformation 1.2 Generalized Node (GN) Conventional nodes (1) Rigid body translations. (2) Rotations around node. (3) Simple tension/compression. (4) Pure shearing. A generalized node New nodal d.o.fs Nodal displacements R.Tian

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