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A Second Order Virtual Node Algorithm for Stokes Flow Problems with Interfacial Forces and Discontinuous Material Properties
"... We present a numerical method for the solution of the Stokes equations that handles interfacial discontinuities due to both singular forces and discontinuous fluid properties such as viscosity and density. The discretization couples a Lagrangian representation of the material interface with an Euler ..."
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We present a numerical method for the solution of the Stokes equations that handles interfacial discontinuities due to both singular forces and discontinuous fluid properties such as viscosity and density. The discretization couples a Lagrangian representation of the material interface with an Eulerian representation of the fluid velocity and pressure. The method is efficient, easy to implement and yields discretely divergencefree velocities that are second order accurate. No knowledge of the jumps on the fluid variables and their derivatives is required along the interface. We discretize the equations using an embedded approach on a uniform MAC grid employing virtual nodes and duplicated cells at the interfaces. These additional degrees of freedom allow for accurate resolution of discontinuities in the fluid stress at the material interface but require a Lagrange multiplier term to enforce continuity of the fluid velocity. We provide a novel discretization of this term that accurately resolves the constant pressure null modes. We show that the accurate resolution of these modes accelerates the overall speed of our simulations. Interfaces are represented with a hybrid Lagrangian/level set method. The discrete coupled equations for the velocity, pressure and Lagrange multipliers are in the form of a symmetric KKT system. Numerical results indicate second order accuracy for the velocities and first order accuracy for the pressure (in L ∞).
High Resolution Sharp Computational Methods for Elliptic and Parabolic Problems in Complex Geometries
, 2012
"... In honor of Stan Osher’s 70 th birthday We present a review of some of the stateoftheart numerical methods for solving the Stefan problem and the Poisson and the diffusion equations on irregular domains using (i) the levelset method for representing the (possibly moving) irregular domain’s bound ..."
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In honor of Stan Osher’s 70 th birthday We present a review of some of the stateoftheart numerical methods for solving the Stefan problem and the Poisson and the diffusion equations on irregular domains using (i) the levelset method for representing the (possibly moving) irregular domain’s boundary, (ii) the ghostfluid method for imposing the Dirichlet boundary condition at the irregular domain’s boundary and (iii) a quadtree/octree nodebased adaptive mesh refinement for capturing small length scales while significantly reducing the memory and CPU footprint. In addition, we highlight common misconceptions and describe how to properly implement these methods. Numerical experiments illustrate quantitative and qualitative results. 1
A Level Set Method for Ductile Fracture
"... We utilize the shape derivative of the classical Griffith’s energy in a level set method for the simulation of dynamic ductile fracture. The level set is defined in the undeformed configuration of the object, and its evolution is designed to represent a transition from undamaged to failed material. ..."
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We utilize the shape derivative of the classical Griffith’s energy in a level set method for the simulation of dynamic ductile fracture. The level set is defined in the undeformed configuration of the object, and its evolution is designed to represent a transition from undamaged to failed material. No remeshing is needed since the resulting topological changes are handled naturally by the level set method. We provide a new mechanism for the generation of fragments of material during the progression of the level set in the Griffith’s energy minimization. Collisions between different material pieces are resolved with impulses derived from the material point method over a background Eulerian grid. This provides a stable means for colliding with embedded interfaces. Simulation of corotational elasticity is based on an implicit finite element discretization.
LargeScale Liquid Simulation on Adaptive Hexahedral Grids
"... Abstract—Regular grids are attractive for numerical fluid simulations because they give rise to efficient computational kernels. However, for simulating high resolution effects in complicated domains they are only of limited suitability due to memory constraints. In this paper we present a method fo ..."
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Abstract—Regular grids are attractive for numerical fluid simulations because they give rise to efficient computational kernels. However, for simulating high resolution effects in complicated domains they are only of limited suitability due to memory constraints. In this paper we present a method for liquid simulation on an adaptive octree grid using a hexahedral finite element discretization, which reduces memory requirements by coarsening the elements in the interior of the liquid body. To impose free surface boundary conditions with second order accuracy, we incorporate a particular class of Nitsche methods enforcing the Dirichlet boundary conditions for the pressure in a variational sense. We then show how to construct a multigrid hierarchy from the adaptive octree grid, so that a time efficient geometric multigrid solver can be used. To improve solver convergence, we propose a special treatment of liquid boundaries via composite finite elements at coarser scales. We demonstrate the effectiveness of our method for liquid simulations that would require hundreds of millions of simulation elements in a nonadaptive regime. Index Terms—Fluid simulation, finite elements, octree, multigrid F 1
Contents lists available at SciVerse ScienceDirect Journal of Computational Physics
"... journal homepage: www.elsevier.com/locate/jcp A second order virtual node algorithm for Stokes flow problems with interfacial forces, discontinuous material ..."
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journal homepage: www.elsevier.com/locate/jcp A second order virtual node algorithm for Stokes flow problems with interfacial forces, discontinuous material
ANALYSIS OF THE DIFFUSEDOMAIN METHOD FOR SOLVING PDES IN COMPLEX GEOMETRIES∗
"... Abstract. In recent work, Li et al. (Comm. Math. Sci., 7:81107, 2009) developed a diffusedomain method (DDM) for solving partial differential equations in complex, dynamic geometries with Dirichlet, Neumann, and Robin boundary conditions. The diffusedomain method uses an implicit representation o ..."
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Abstract. In recent work, Li et al. (Comm. Math. Sci., 7:81107, 2009) developed a diffusedomain method (DDM) for solving partial differential equations in complex, dynamic geometries with Dirichlet, Neumann, and Robin boundary conditions. The diffusedomain 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 diffusedomain methods for Neumann and Robin boundary conditions. Our analysis shows that for certain choices of the boundary condition approximations, the DDM is secondorder accurate in . However, for other choices the DDM is only firstorder accurate. This helps to explain why the choice of boundarycondition approximation is important for rapid global convergence and high accuracy. Our analysis also suggests correction terms that may be added to yield more accurate diffusedomain methods. Simple modifications of firstorder boundary condition approximations are proposed to achieve asymptotically secondorder accurate schemes. Our analytic results are confirmed numerically in the L2 and L ∞ norms for selected test problems. Key words. numerical solution of partial differential equations, phasefield approximation, implicit geometry representation, matched asymptotic analysis.
A Second Order Virtual Node Algorithm for NavierStokes Flow Problems with Interfacial Forces and Discontinuous Material Properties
"... We present a numerical method for the solution of the NavierStokes equations in three dimensions that handles interfacial discontinuities due to singular forces and discontinuous fluid properties such as viscosity and density. We show that this also allows for the enforcement of normal stress and v ..."
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We present a numerical method for the solution of the NavierStokes equations in three dimensions that handles interfacial discontinuities due to singular forces and discontinuous fluid properties such as viscosity and density. We show that this also allows for the enforcement of normal stress and velocity boundary conditions on irregular domains. The method improves on results in [1] (which solved the Stokes equations in two dimensions) by providing treatment of fluid inertia as well as a new discretization of jump and boundary conditions that accurately resolves null modes in both two and three dimensions. We discretize the equations using an embedded approach on a uniform MAC grid to yield discretely divergencefree velocities that are second order accurate. We maintain our interface using the level set method or, when more appropriate, the particle level set method. We show how to implement Dirichlet (known velocity), Neumann (known normal stress), and slip velocity boundary conditions as special cases of our interface representation. The method leads to a discrete, symmetric KKT system for velocities, pressures, and Lagrange multipliers. We also present a novel simplification to the standard combination of the second order semiLagrangian and BDF schemes for discretizing the inertial terms. Numerical results indicate second order spatial accuracy for the velocities (L ∞ and L2) and first order for the pressure (in L∞, second order in L2). Our temporal discretization is also second order accurate.