Smooth Two-Dimensional Interpolations: A Recipe for All Polygons

Malsch, Elisabeth; Lin, John Jeffy; Dasgupta, Gautam

doi:10.1080/2151237x.2005.10129192

Cited by 38 publications

(26 citation statements)

References 12 publications

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“…The interpolant is a function of Lebesgue measure of Voronoi edge and L 2 distance norm Mean value coordinates [12,13] Yes Interpolant is a function of geometric quantities -L 2 distance norm and area Metric coordinate method [14] Yes Uses measures such as edge length, signed area of triangle, and trigonometric functions of sine and cosine to construct the shape functions Maximum entropy [15][16][17] Yes Shape functions and their derivatives are obtained by maximizing the Shannon's entropy function under prescribed boundary conditions Harmonic coordinates [20][21][22] Yes Shape functions and their derivatives are obtained by solving the Laplace equation hierarchically a b which is computationally expensive. Moreover, to achieve accurate results, a high order numerical quadrature is required.…”

Section: Interpolantsmentioning

confidence: 99%

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Topology optimization using polytopes

Gain

Paulino

Duarte

et al. 2015

Computer Methods in Applied Mechanics and Engineering

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Meshing complex engineering domains is a challenging task. Arbitrary polyhedral meshes can provide the much needed flexibility in automated discretization of such domains. The geometric property of polyhedral meshes such as its unstructured nature and the connectivity of faces between elements makes them specially attractive for topology optimization applications. Numerical anomalies in designs such as the single node connections and checkerboard pattern can be naturally circumvented with polyhedrons. In the current work, we solve the governing three-dimensional elasticity state equation using the Virtual Element Method (VEM) approach. The main characteristic difference between VEM and standard finite element methods (FEM) is that in VEM the canonical basis functions are not constructed explicitly. Rather the stiffness matrix is computed directly utilizing a projection map which extracts the linear component of the deformation. Such a construction guarantees the satisfaction of the patch test (used by engineers as an indicator of optimal convergence of numerical solutions under mesh refinement). Finally, the computations reduce to the evaluation of matrices which contain purely geometric surface facet quantities. The present work focuses on the first-order VEM in which the degrees of freedom are associated with the vertices. The features of the current optimization approach are demonstrated using numerical examples for compliance minimization and compliant mechanism problems.

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Section: Interpolantsmentioning

confidence: 99%

“…Based on the kinematic decomposition of deformation state (14) and the energy orthogonality property (12), the continuous bilinear form can be written as:…”

Section: Theoretical Backgroundmentioning

confidence: 99%

Topology optimization using polytopes

Gain

Paulino

Duarte

et al. 2015

Computer Methods in Applied Mechanics and Engineering

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“…An alternative approach to derive the polygonal interpolants was proposed by Malsch et al (2005) by employing the metric coordinate system that can be used in convex or concave polygons, or any polygonal domain containing isolated points in its interior. The resulting shape functions are defined at any point of the polygon that satisfy the boundedness, completeness and linearity on each side.…”

Section: Metric Coordinatesmentioning

confidence: 99%

A polygonal finite element method for modeling crack propagation with minimum remeshing

2015

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In this paper, a polygonal finite element method is presented for crack growth simulation with minimum remeshing. A local polygonal mesh strategy is performed employing polygonal finite element method to model the crack propagation. In order to model the singular crack tip fields, the convex and concave polygonal elements are modified based on the singular quarter point isoparametric concept that improves the accuracy of the stress intensity factors. Numerical simulations are performed to demonstrate the efficiency of various polygonal shape functions, including the Wachspress, metric, Laplace and mean value shape functions, in modeling the crack tip fields. Eventually, analogy of the results with the existing numerical and experimental data is carried out depicting accuracy of the propounded technique.

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“…5). These properties are not met by other barycentric constructions [13,16]. Hormann and Floater [15] have presented the important properties of mean value coordinates along with an implementation for planar polygons.…”

Section: Polygonal Meshesmentioning

confidence: 99%

“…Many new contributions on barycentric polygonal interpolation have been realized in geometry modeling and graphics, and in finite element methods [1,[9][10][11][12][13][14][15][16]. In Reference [1], natural neighbor based (Laplace) interpolants [17] are used to construct shape functions on irregular polygons (n-gons), which is adapted to quadtree meshes (resolves the issue of hanging nodes) in References [18,19].…”

Section: Introductionmentioning

confidence: 99%

Extended finite element method on polygonal and quadtree meshes

Tabarraei

Sukumar

2008

Computer Methods in Applied Mechanics and Engineering

139

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In this paper, we present mesh-independent modeling of discontinuous fields on polygonal and quadtree finite element meshes. This approach falls within the class of extended and generalized finite element methods, where the partition of unity framework is used to introduce additional (enrichment) functions within the classical displacement-based finite element approximation. For crack modeling, a discontinuous function and the two-dimensional asymptotic crack-tip fields are used as enrichment functions. Linearly complete partition of unity approximations are adopted on polygonal (convex and nonconvex elements) and quadtree meshes. Excellent agreement with reference solution results is obtained for mixed-mode stress intensity factors on benchmark crack problems, and crack growth simulations without remeshing are conducted on polygonal and quadtree meshes to reveal the potential of the proposed techniques in computational failure mechanics.

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Smooth Two-Dimensional Interpolations: A Recipe for All Polygons

Cited by 38 publications

References 12 publications

Topology optimization using polytopes

Topology optimization using polytopes

A polygonal finite element method for modeling crack propagation with minimum remeshing

Extended finite element method on polygonal and quadtree meshes

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