Barycentric Coordinates and Their Properties

Anisimov, Dmitry

doi:10.1201/9781315153452-1

Cited by 8 publications

(7 citation statements)

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“…This has connections to generalized barycentric coordinates and in particular to mean value interpolation over polygons and curved domains [75]. Generalized barycentric coordinates [76][77][78] are an extension of barycentric coordinates over simplices to polygons and polyhedra. These coordinates (shape functions) have linear precision and are nonnegative over convex polygons.…”

Section: Generalized Mean Value Potentials and L P -Distance Fieldsmentioning

confidence: 99%

“…Consider a planar polygon with n vertices (nodal coordinate {x i } n i=1 ) that are in counterclockwise orientation. On an n-gon, harmonic coordinates [95] are one of the instances of generalized barycentric coordinates [77]. Each coordinate (shape function), ϕ i ≡ ϕ i (x), is associated with vertex i and is obtained by solving the Laplace equation with piecewise affine Dirichlet boundary conditions.…”

Section: Generalized Barycentric Coordinates Over Polygonsmentioning

confidence: 99%

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Exact imposition of boundary conditions with distance functions in physics-informed deep neural networks

Sukumar,

Srivastava

2021

Preprint

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In this paper, we introduce a new approach based on distance fields to exactly impose boundary conditions in physicsinformed deep neural networks. The challenges in satisfying Dirichlet boundary conditions in meshfree and particle methods are well-known. This issue is also pertinent in the development of physics informed neural networks (PINN) for the solution of partial differential equations. We introduce geometry-aware trial functions in artifical neural networks to improve the training in deep learning for partial differential equations. To this end, we use concepts from constructive solid geometry (R-functions) and generalized barycentric coordinates (mean value potential fields) to construct φ(x), an approximate distance function to the boundary of a domain in R d . To exactly impose homogeneous Dirichlet boundary conditions, the trial function is taken as φ(x) multiplied by the PINN approximation, and its generalization via transfinite interpolation is used to a priori satisfy inhomogeneous Dirichlet (essential), Neumann (natural), and Robin boundary conditions on complex geometries. In doing so, we eliminate modeling error associated with the satisfaction of boundary conditions in a collocation method and ensure that kinematic admissibility is met pointwise in a Ritz method. With this new ansatz, the training for the neural network is simplified: sole contribution to the loss function is from the residual error at interior collocation points where the governing equation is required to be satisfied. Numerical solutions are computed using strong form collocation and Ritz minimization. To convey the main ideas and to assess the accuracy of the approach, we present numerical solutions for linear and nonlinear boundary-value problems over convex and nonconvex polygonal domains as well as over domains with curved boundaries. Benchmark problems in one dimension for linear elasticity, advection-diffusion, and beam bending; and in two dimensions for the steady-state heat equation, Laplace equation, biharmonic equation (Kirchhoff plate bending), and the nonlinear Eikonal equation are considered. The construction of approximate distance functions using R-functions extends to higher dimensions, and we showcase its use by solving a Poisson problem with homogeneneous Dirichlet boundary conditions over the four-dimensional hypercube. The proposed approach consistently outperforms a standard PINN-based collocation method, which underscores the importance of exactly (a priori) satisfying the boundary condition when constructing a loss function in PINN. This study provides a pathway for meshfree analysis to be conducted on the exact geometry without domain discretization.

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Section: Generalized Mean Value Potentials and L P -Distance Fieldsmentioning

confidence: 99%

Section: Generalized Barycentric Coordinates Over Polygonsmentioning

confidence: 99%

Exact imposition of boundary conditions with distance functions in physics-informed deep neural networks

Sukumar,

Srivastava

2021

Preprint

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“…For n ≥ 4, λ k ( x ) is no longer unique. By adopting different definitions of w k ( x ) using length and area metrics, various generalized barycentric coordinates have been developed, 50,51 with mean value coordinates being widely adopted because they are well‐defined for star‐shaped and arbitrary polygons 50 …”

Section: Proposed Displacement Function For Ddamentioning

confidence: 99%

“…Generalized barycentric coordinates, which are typically used in computer graphics, have been adopted as suitable shape function for polygonal finite elements, as demonstrated in References 48,49. Various types of generalized barycentric coordinates have been developed, as summarized until 2016 by Floater 50 and Anisimov 51 …”

Section: Introductionmentioning

confidence: 99%

Novel displacement function for discontinuous deformation analysis based on mean value coordinates

Jiang

Zheng

et al. 2020

Numerical Meth Engineering

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The rotation degree of freedom in discontinuous deformation analysis (DDA) may cause false volume expansion when a block undergoes a large rotation. We propose a block displacement function to prevent this defect. Specifically, the degrees of freedom in a block are redefined by incremental displacements at its vertices, and displacement is formulated based on the mean value coordinates. In addition, the finite element method with updated Lagrangian formulation is employed to derive the equilibrium equations, while the contact analysis and implicit time integration for dynamics is maintained from the original DDA. After each time step, the block configuration is updated by adding the new degrees of freedom to the previous coordinates of the block vertices. Results from numerical examples confirm the effectiveness of the proposed approach to prevent false volume expansion, ensure correctness of contact analysis, and provide realistic stress results when simulating large rotations. K E Y W O R D S discontinuous deformation analysis (DDA), false volume expansion, mean value coordinates 1 INTRODUCTION Discontinuous deformation analysis (DDA), originally proposed by Shi, 1 allows to effectively analyze the mechanical behavior of a discrete block system subject to large displacements. 2 In addition, DDA can handle discontinuity and has been adopted to address diverse geotechnical problems, such as landslide simulations, 3,4 rock slope stability evaluation, 5,6 rockfall trajectory tracking, 7 blasting effect evaluation, 8 shear creep deformation analysis of structural planes, 9 crack propagation modeling, 10,11 and seismic wave propagation modeling. 12 Various enhancements have been 4768

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“…Consider an open, bounded convex domain Ω with boundary Γ = ∂Ω. Generalized barycentric coordinates [53,54], φ : Ω → R m + , on an m-gon are nonnegative and satisfy the partition of unity and linear precision properties:…”

Section: Transfinite Mean Value Interpolationmentioning

confidence: 99%

Scaled boundary cubature scheme for numerical integration over planar regions with affine and curved boundaries

Chin,

Sukumar

2020

Preprint

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This paper introduces the scaled boundary cubature (SBC) scheme for accurate and efficient integration of functions over polygons and two-dimensional regions bounded by parametric curves. A forthcoming paper will cover integration over polyhedra and solids bounded by curved surfaces using the SBC scheme. Over two-dimensional domains, the SBC method reduces integration over a region bounded by m curves to integration over m regions (referred to as curved triangular regions), where each region is bounded by two line segments and a curve. With proper (counterclockwise) orientation of the boundary curves, the scheme is applicable to convex and nonconvex domains. Additionally, for star-convex domains, a tensor-product cubature rule with positive weights and integration points in the interior of the domain is obtained. If the integrand is homogeneous, we show that this new method reduces to the homogeneous numerical integration scheme [1, 2]; however, the SBC scheme is more versatile since it is equally applicable to both homogeneous and non-homogeneous functions. This paper also introduces several methods for smoothing integrands with point singularities and near-singularities based on the generalized Duffy transformation [3] and a distance transformation described in Ma and Kamiya [4]. When these methods are combined, highly efficient integration of weakly-singular functions is realized. The SBC method is applied to a number of benchmark problems, which reveal its broad applicability and superior performance (in terms of time to generate a rule and accuracy per cubature point) when compared to existing methods for integration.

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Barycentric Coordinates and Their Properties

Cited by 8 publications

References 0 publications

Exact imposition of boundary conditions with distance functions in physics-informed deep neural networks

Exact imposition of boundary conditions with distance functions in physics-informed deep neural networks

Novel displacement function for discontinuous deformation analysis based on mean value coordinates

Scaled boundary cubature scheme for numerical integration over planar regions with affine and curved boundaries

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