A second-order virtual node algorithm for nearly incompressible linear elasticity in irregular domains

Zhu, Yongning; Wang, Yuting; Hellrung, Jeffrey Lee; Cantarero, Alejandro; Sifakis, Eftychios; Teran, Joseph

doi:10.1016/j.jcp.2012.05.015

Cited by 7 publications

(8 citation statements)

References 63 publications

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“…However, most of these methods typically require tools not frequently available in standard finite element and finite difference software packages. Examples of such approaches include the extended and composite finite element methods (e.g., [31,12,23,13,32,55,7,4]), immersed interface methods (e.g., [40,43,60,44,65]), virtual node methods with embedded boundary conditions (e.g., [3,73,34]), matched interface and boundary methods (e.g., [71,68,69,67,72]), modified finite volume/embedded boundary/cut-cell methods/ghost-fluid methods (e.g., [27,36,19,25,26,35,47,70,48,37,46,64,49,9,10,52,53,33,63]). In another approach, known as the fictitious domain method (e.g., [28,29,56,45]), the original system is either augmented with equations for Lagrange multipliers to enforce the boundary conditions, or the penalty method is used to enforce the boundary condi-tions weakly.…”

Section: Introductionmentioning

confidence: 99%

Analysis of the diffuse-domain method for solving PDEs in complex geometries

Lervåg

Lowengrub

2015

Communications in Mathematical Sciences

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Abstract. In recent work, Li et al. (Comm. Math. Sci., 7:81-107, 2009) developed a diffusedomain method (DDM) for solving partial differential equations in complex, dynamic geometries with Dirichlet, Neumann, and Robin boundary conditions. The diffuse-domain method uses an implicit representation of the geometry where the sharp boundary is replaced by a diffuse layer with thickness that is typically proportional to the minimum grid size. The original equations are reformulated on a larger regular domain and the boundary conditions are incorporated via singular source terms. The resulting equations can be solved with standard finite difference and finite element software packages.Here, we present a matched asymptotic analysis of general diffuse-domain methods for Neumann and Robin boundary conditions. Our analysis shows that for certain choices of the boundary condition approximations, the DDM is second-order accurate in . However, for other choices the DDM is only first-order accurate. This helps to explain why the choice of boundary-condition approximation is important for rapid global convergence and high accuracy. Our analysis also suggests correction terms that may be added to yield more accurate diffuse-domain methods. Simple modifications of first-order boundary condition approximations are proposed to achieve asymptotically second-order accurate schemes. Our analytic results are confirmed numerically in the L 2 and L ∞ norms for selected test problems.

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Section: Introductionmentioning

confidence: 99%

Analysis of the diffuse-domain method for solving PDEs in complex geometries

Lervåg

Lowengrub

2015

Communications in Mathematical Sciences

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“…Explicit boundary discretizations, such as embedded surface meshes, show their unique merits in modeling the sub-cell geometry and enforcing precise boundary conditions [Azevedo et al 2016;Schroeder et al 2012]. These embedded discretizations of the variational type, are focused on handling Dirichlet boundaries Zhu et al 2012], which inspired our discretization for solving steady-state flow problems. These discretizations can conveniently accommodate adaptive resolution [Ando et al 2013] and flows in containers with deforming geometry [Feldman et al 2005].…”

Section: Related Workmentioning

confidence: 99%

“…Our model: quasi-incompressible Stokes. The aforementioned relation of Stokes and linear elasticity has previously been leveraged primarily to develop discretization and solution schemes for incompressible or near-incompressible elasticity that draw inspiration from established methods for Stokes [Gaspar et al 2008;Zhu et al 2012]. However, directly pursuing a discretization of the Stokes problem has its own subtleties; due to the incompressibility constraint, the associated discretizations -and especially variational formulations -take the form of saddle point problems, restricting somewhat the options for associated numerical solvers.…”

Section: Governing Partial Differential Equationsmentioning

confidence: 99%

“…It is known [Daux et al 2000;Hughes 2012] that the associated variational formulation of this problem computes the solution via minimization of an energy functional [ ] over all functions in an appropriate solution space. For our purposes, we define the solution space to be all functions defined by bilinear (2D) or trilinear (3D) interpolation over the cells of the background Cartesian grid, and the associated energy to be minimized is [Daux et al 2000;Zhu et al 2012]…”

Section: Numerical Discretizationmentioning

confidence: 99%

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Functional optimization of fluidic devices with differentiable stokes flow

Du¹,

Wu²,

Spielberg³

et al. 2020

ACM Trans. Graph.

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We present a method for performance-driven optimization of fluidic devices. In our approach, engineers provide a high-level specification of a device using parametric surfaces for the fluid-solid boundaries. They also specify desired flow properties for inlets and outlets of the device. Our computational approach optimizes the boundary of the fluidic device such that its steady-state flow matches desired flow at outlets. In order to deal with computational challenges of this task, we propose an efficient, differentiable Stokes flow solver. Our solver provides explicit access to gradients of performance metrics with respect to the parametric boundary representation. This key feature allows us to couple the solver with efficient gradient-based optimization methods. We demonstrate the efficacy of this approach on designs of five complex 3D fluidic systems. Our approach makes an important step towards practical computational design tools for high-performance fluidic devices.

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“…They generate an explicit discontinuity-aligned boundary mesh and compute surface integrals using the divergence-theorem. VNA-based approaches following the same strategy were also developed in the engineering community [Bedrossian et al 2010;Zhu et al 2012]. Over the years several approaches based on equivalent polynomials [Ventura 2006], adaptive quadrature [Müller et al 2012] or variable quadrature weights [Holdych et al 2008] were developed.…”

Section: Related Workmentioning

confidence: 99%

Robust eXtended finite elements for complex cutting of deformables

2017

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In this paper we present a robust remeshing-free cutting algorithm on the basis of the eXtended Finite Element Method (XFEM) and fully implicit time integration. One of the most crucial points of the XFEM is that integrals over discontinuous polynomials have to be computed on subdomains of the polyhedral elements. Most existing approaches construct a cut-aligned auxiliary mesh for integration. In contrast, we propose a cutting algorithm that includes the construction of specialized quadrature rules for each dissected element without the requirement to explicitly represent the arising subdomains. Moreover, we solve the problem of ill-conditioned or even numerically singular solver matrices during time integration using a novel algorithm that constrains non-contributing degrees of freedom (DOFs) and introduce a preconditioner that efficiently reuses the constructed quadrature weights. Our method is particularly suitable for fine structural cutting as it decouples the added number of DOFs from the cut's geometry and correctly preserves geometry and physical properties by accurate integration. Due to the implicit time integration these fine features can still be simulated robustly using large time steps. As opposed to this, the vast majority of existing approaches either use remeshing or element duplication. Remeshing based methods are able to correctly preserve physical quantities but strongly couple cut geometry and mesh resolution leading to an unnecessary large number of additional DOFs. Element duplication based approaches keep the number of additional DOFs small but fail at correct conservation of mass and stiffness properties. We verify consistency and robustness of our approach on simple and reproducible academic examples while stability and applicability are demonstrated in large scenarios with complex and fine structural cutting.

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A second-order virtual node algorithm for nearly incompressible linear elasticity in irregular domains

Cited by 7 publications

References 63 publications

Analysis of the diffuse-domain method for solving PDEs in complex geometries

Analysis of the diffuse-domain method for solving PDEs in complex geometries

Functional optimization of fluidic devices with differentiable stokes flow

Robust eXtended finite elements for complex cutting of deformables

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