THE VIRTUAL ELEMENT DECOMPOSITION: A NEW PARADIGM … · Developed NIVED (Nodal Integration –VED)...
Transcript of THE VIRTUAL ELEMENT DECOMPOSITION: A NEW PARADIGM … · Developed NIVED (Nodal Integration –VED)...
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WONAPDE 2019Sixth Chilean Workshop on Numerical Analysis of Partial Differential Equations
Concepción Chile January 22, 2019
THE VIRTUAL ELEMENT DECOMPOSITION: A NEW PARADIGM FOR DEVELOPING NODAL INTEGRATION
SCHEMES FOR MESHFREE GALERKIN METHODS
Rodrigo Silva-Valenzuela, Alejandro Ortiz-BernardinDepartment of Mechanical Engineering
Universidad de Chilewww.camlab.cl/alejandro
Joint work with:N. Sukumar (UC-Davis)E. Artioli (U. Rome-Tor Vergata)
Funded through grant CONICYT/FONDECYT No. 1181192
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Motivation
Meshfree Basis Functions
Meshfree Integration Cells
Nodal Integration Using the Virtual Element Decomposition (NIVED)
Numerical Tests
Summary and Outlook
Outline
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Motivation
Numerical integration of the stiffness matrix
Gauss integration Nodal integration Displacements at nodes Stresses/Strains at Gauss points
Displacements at nodes Stresses/Strains at nodes
Direct integration at nodes leads to instabilities A penalty stabilization is added to the stiffness matrix
Puso et al., IJNME, 74(3), 2008:
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Rational (nonpolynomial) basis functions Linear approximation plus some nonpolynomial terms Integration errors affecting consistency and stability
Meshfree Basis Functions
Belytschko et al., IJNME, 37(2), 1994; Liu et al., IJNMF, 20(8-9), 1995; Atluri & Zhu, CM, 22(2), 1998; De & Bathe, CM, 25(4), 2000;
Sukumar et al., IJNME, 43(5), 1998; Sukumar, IJNME, 61(12), 2004; Arroyo & Ortiz, IJNME, 65(13), 2006… and many others!
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Gauss Integration
Nodal Integration
Meshfree Integration Cells
(Chen et al., IJNME, 50(2), 2001;Chen et al., IJNME, 53(12), 2002; Puso et al., IJNME, 74(3), 2008;Chen et al., IJNME, 95(5), 2013)
(Dolbow & Belytschko, CM, 23(3), 1999;Babŭska et al., IJNME, 76(9), 2008; Ortiz et al., CMAME, 199(25-28), 2010;Duan et al., IJNME, 92(4), 2012;Chen et al., IJNME, 95(5), 2013;Ortiz-Bernardin et al., IJNME, 112(7), 2017)
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Bilinear form (linear elasticity) at the cell level:
Approximation basis:
Displacement spaces:linear displacementsnonpolynomial displacements
Adapted from (O-B et al., IJNME, 112(7), 2017)
NIVED: Basis and Spaces
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Let . Then,
First term gives consistency and second one stability. The stability term can be approximated, which gives
The projection satisfies the orthogonality condition
which allows writing
NIVED: Virtual Element Decomp.
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Using the “intrinsic” formula given in (BdV et al., M3A, 24(8), 2014), the projection operator sampled at is
where
NIVED: Projection Operator
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Nodal integration: sampling the consistency part at and using the projection operator leads to
(consistency stiffness)
Virtual element decomposition:
consistency stability
NIVED: Cell Stiffness Matrix
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For the stability part choose such that it is positive definite and scales uniformly with the exact bilinear form (e.g.: Gain et al., CMAME, 282, 2014):
where is an scaling parameter.
(stability stiffness)
Virtual element decomposition:
consistency stability
NIVED: Cell Stiffness Matrix
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cloud of nodes (including ) from to is the integration point with coordinates is the integration cell and the representative nodal volume X are auxiliary integration points on the boundary of the cell Contribution: support of touches the cell
NIVED: Implementation
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Everything reduces to the computation of the constant strain over the element:
By applying the divergence theorem, these integrals are evaluated on the boundary of the cell:
NIVED: Implementation (Cont’d)
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NIVED
X
Meshes use the same node set
1-pt per edge(basis function only)
MEMInternal Gauss points:1-pt, 3-pt, 6-pt, 12-pt
(basis functions and derivatives)
Numerical Tests: Integration Cells
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Test: Patch Test
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Convergence
coarsest finest
Test: Cantilever Beam
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coarsest finest
Test: Plate With a HoleConvergence
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coarsest finest
Body force: Dirichlet BCs:
Test: Manufactured ProblemConvergence
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Test: Manufactured Problem (Cont’d)Timing
Normalized CPU time =
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Summary and Outlook
Developed NIVED (Nodal Integration – VED) for linear elasticity
VED provides consistency (patch test satisfaction) and optimal convergence to the nodal integration method
Computational cost: NIVED similar to 3-pt Gauss rule
NIVED for nonlinear problems (elastoplasticity) is under development
Outlook: large deformations, 3D simulations
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Journal Acronyms
• M3A: Mathematical Models and Methods in Applied Sciences
• CMAME: Computer Methods in Applied Mechanics and Engineering
• IJNME: International Journal for Numerical Methods in Engineering
• IJNMF: International Journal for Numerical Methods in Fluids
• CM: Computational Mechanics