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Ashok Kumar - One of the best experts on this subject based on the ideXlab platform.
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Isoparametric B-Spline Elements for Immersed Boundary Explicit Dynamic Simulation
Journal of Computing and Information Science in Engineering, 2020Co-Authors: Shashank Menon, Ashok KumarAbstract:Abstract Explicit dynamic analysis has proven to be advantageous when simulating shock and impact loading, and very small time-scale events. In this article, the feasibility of using a Background Mesh of B-spline elements for immersed boundary explicit dynamic simulation is studied. In this approach, the geometry is immersed in a Background Mesh consisting of uniform regular shaped elements to avoid Mesh generation difficulties. The boundary conditions are applied using the step boundary method, which uses the equations of the boundaries to construct trial functions that satisfy the essential boundary conditions. An isoparametric formulation is presented for quadratic and cubic B-spline elements and their shape functions are derived from the classical recursive definition of B-splines. The effectiveness of mass diagonalization for B-spline elements is also explored. This approach is validated using several examples by comparing with modal superposition solutions as well as past work using traditional finite element analysis (FEA) and analytical solutions when available.
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Survey of Immersed Boundary Approaches for Finite Element Analysis
Journal of Computing and Information Science in Engineering, 2020Co-Authors: Ashok KumarAbstract:Abstract Mesh generation for traditional finite element analysis has proven to be very difficult to fully automate especially using hexahedral elements for complex 3D geometry. Several modifications to the finite element method (FEM), such as the Meshless methods, have been proposed for avoiding Mesh generation. An alternative approach has recently gained popularity where the geometry, created as a solid model in cad software, is embedded or immersed in a nonconforming Background Mesh for analysis. In this approach, referred to here as the immersed boundary approach, a Background Mesh that is independent of the geometry is used for piecewise interpolation or approximation of the solution. Therefore, a uniform Mesh with regular-shaped or undistorted elements can be used, and such a Mesh is easy to generate automatically. When the geometry is immersed in the Background Mesh, the boundary elements are often only partly inside the geometry and the nodes of the Mesh may not be on the boundaries. Many new methods have been developed to integrate over partial elements and to apply boundary and interface conditions when the boundaries of the geometries do not conform to the Background Mesh. These methods are reviewed in this article with particular emphasis on the implicit boundary method and step boundary method for applying boundary conditions. In addition, B-spline elements and several applications of the immersed boundary approach are surveyed including composite microstructures and structural elements for plates and shells.
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total lagrangian formulation for large deformation modeling using uniform Background Mesh
ASME 2016 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, 2016Co-Authors: Nikhil Bhosale, Ashok KumarAbstract:Mesh generation difficulties can be avoided when a Background Mesh rather than a Mesh that conforms to the geometry is used for the analysis. The geometry is represented by equations and is independent of the Mesh and is immersed in the Background Mesh. The solution to boundary value problems is approximated or piece-wise interpolated using the Background Mesh. The main challenge is in applying the boundary conditions because the boundaries may not have any nodes on them. Implicit boundary method has been used for linear static and dynamic analysis and has shown to be an effective approach for imposing boundary conditions but has never been applied to nonlinear problems. In this paper, this approach is extended to large deformation nonlinear analysis using the Total Lagrangian formulation. The equations are solved using the widely used modified Newton-Raphson method with loads applied over many load steps. Several test examples are studied and compared with traditional finite element analysis software for verification.Copyright © 2016 by ASME
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Coupled Electrostatic-Structural Analysis Using Mesh Independent Approach
Volume 1A: 35th Computers and Information in Engineering Conference, 2015Co-Authors: Pushkar Mishra, Ashok KumarAbstract:Coupled electrostatic-structural interaction needs to be modeled for designing many electrostatically actuated devices such as MEMS switches, accelerometers and comb-drives. The models for such devices must include an assembly of components and the air gaps between them. The electrostatic fields are generated in the air gaps and this field produces electrostatic forces which act on the structural components causing deformation and strains. Generating an appropriate Mesh for such an assembly of regions with different material properties can be cumbersome. In this paper, we have developed an immersed boundary approach where the geometry is defined using the equations of the boundary and is immersed in a uniform Background Mesh. The loads and boundary conditions are imposed using the implicit boundary method since the nodes of the Background Mesh are not guaranteed to be on the boundaries. This approach is verified by applying it to commonly used examples.Copyright © 2015 by ASME
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Modal Analysis Using Implicit Boundary Finite Element Methods
Volume 1A: 34th Computers and Information in Engineering Conference, 2014Co-Authors: Zhiyuan Zhang, Ashok KumarAbstract:Modal analysis is widely used for linear dynamic analysis of structures. The finite element method is used to numerically compute stiffness and mass matrices and the corresponding eigenvalue problem is solved to determine the natural frequencies and mode shapes of vibration. Implicit boundary method was developed to use equations of the boundary to apply boundary conditions and loads so that a Background Mesh can be used for analysis. A Background Mesh is easier to generate because the elements do not have to conform to the given geometry and therefore uniform regular shaped elements can be used. In this paper, we show that this approach is suitable for modal analysis and modal superposition techniques as well. Furthermore, the implicit boundary method also allows higher order elements that use B-spline approximations. Several test examples are studied for verification.Copyright © 2014 by ASME
Hailong Chen - One of the best experts on this subject based on the ideXlab platform.
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method for imposing boundary conditions on reissner mindlin plates for analysis using structured Background Mesh
Computers & Structures, 2014Co-Authors: Hailong Chen, Ashok KumarAbstract:Elements in a conforming Mesh are often distorted to fit the geometry which can cause numerical errors. If a Background Mesh is used for analysis, the elements can be regular shaped. The geometry must be represented independently using equations and implicit boundary method can be used to enforce essential boundary conditions. In this paper, this approach is extended to traditional Reissner–Mindlin plate elements that use mixed interpolation to avoid shear locking. The convergence of plate elements that use implicit boundary approach and a Background Mesh is studied in this paper using several benchmark problems to test their validity and robustness.
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Method for imposing boundary conditions on Reissner–Mindlin plates for analysis using structured Background Mesh
Computers & Structures, 2014Co-Authors: Hailong Chen, Ashok KumarAbstract:Elements in a conforming Mesh are often distorted to fit the geometry which can cause numerical errors. If a Background Mesh is used for analysis, the elements can be regular shaped. The geometry must be represented independently using equations and implicit boundary method can be used to enforce essential boundary conditions. In this paper, this approach is extended to traditional Reissner–Mindlin plate elements that use mixed interpolation to avoid shear locking. The convergence of plate elements that use implicit boundary approach and a Background Mesh is studied in this paper using several benchmark problems to test their validity and robustness.
Charbel Farhat - One of the best experts on this subject based on the ideXlab platform.
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Accelerated Mesh sampling for the hyper reduction of nonlinear computational models
International Journal for Numerical Methods in Engineering, 2017Co-Authors: Todd Chapman, Pat Collins, P. Avery, Charbel FarhatAbstract:We describe a method for discretizing planar C2-regular domains immersed in non-conforming triangulations. The method consists in constructing mappings from triangles in a Background Mesh to curvilinear ones that conform exactly to the immersed domain. Constructing such a map relies on a novel way of parameterizing the immersed boundary over a collection of nearby edges with its closest point projection. By interpolating the mappings to curvilinear triangles at select points, we recover isoparametric mappings for the immersed domain defined over the Background Mesh. Indeed, interpolating the constructed mappings just at the vertices of the Background Mesh yields a fast Meshing algorithm that involves only perturbing a few vertices near the boundary. For the discretization of a curved domain to be robust, we have to impose restrictions on the Background Mesh. Conversely, these restrictions define a family of domains that can be discretized with a given Background Mesh. We then say that the Background Mesh is a universal Mesh for such a family of domains. The notion of universal Meshes is particularly useful in free/moving boundary problems because the same Background Mesh can serve as the universal Mesh for the evolving domain for time intervals that are independent of the time step. Hence it facilitates a framework for finite element calculations over evolving domains while using a fixed Background Mesh. Furthermore, since the evolving geometry can be approximated with any desired order, numerical solutions can be computed with high-order accuracy. We demonstrate these ideas with various numerical examples.
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A FETI-preconditioned conjugate gradient method for large-scale stochastic finite element problems
International Journal for Numerical Methods in Engineering, 2009Co-Authors: Debraj Ghosh, P. Avery, Charbel FarhatAbstract:We describe a method for discretizing planar C2-regular domains immersed in non-conforming triangulations. The method consists in constructing mappings from triangles in a Background Mesh to curvilinear ones that conform exactly to the immersed domain. Constructing such a map relies on a novel way of parameterizing the immersed boundary over a collection of nearby edges with its closest point projection. By interpolating the mappings to curvilinear triangles at select points, we recover isoparametric mappings for the immersed domain defined over the Background Mesh. Indeed, interpolating the constructed mappings just at the vertices of the Background Mesh yields a fast Meshing algorithm that involves only perturbing a few vertices near the boundary. For the discretization of a curved domain to be robust, we have to impose restrictions on the Background Mesh. Conversely, these restrictions define a family of domains that can be discretized with a given Background Mesh. We then say that the Background Mesh is a universal Mesh for such a family of domains. The notion of universal Meshes is particularly useful in free/moving boundary problems because the same Background Mesh can serve as the universal Mesh for the evolving domain for time intervals that are independent of the time step. Hence it facilitates a framework for finite element calculations over evolving domains while using a fixed Background Mesh. Furthermore, since the evolving geometry can be approximated with any desired order, numerical solutions can be computed with high-order accuracy. We demonstrate these ideas with various numerical examples.
André Massing - One of the best experts on this subject based on the ideXlab platform.
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Cut finite element methods for partial differential equations on embedded manifolds of arbitrary codimensions
Mathematical Modelling and Numerical Analysis, 2018Co-Authors: Erik Burman, Peter Hansbo, Mats G. Larson, André MassingAbstract:We develop a theoretical framework for the analysis of stabilized cut finite element methods for the Laplace-Beltrami operator on a manifold embedded in ℝd of arbitrary codimension. The method is based on using continuous piecewise linears on a Background Mesh in the embedding space for approximation together with a stabilizing form that ensures that the resulting problem is stable. The discrete manifold is represented using a triangulation which does not match the Background Mesh and does not need to be shape-regular, which includes level set descriptions of codimension one manifolds and the non-matching embedding of independently triangulated manifolds as special cases. We identify abstract key assumptions on the stabilizing form which allow us to prove a bound on the condition number of the stiffness matrix and optimal order a priori estimates. The key assumptions are verified for three different realizations of the stabilizing form including a novel stabilization approach based on penalizing the surface normal gradient on the Background Mesh. Finally, we present numerical results illustrating our results for a curve and a surface embedded in ℝ3 .
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A Cut Discontinuous Galerkin Method for Coupled Bulk-Surface Problems
arXiv: Numerical Analysis, 2017Co-Authors: André MassingAbstract:We develop a cut Discontinuous Galerkin method (cutDGM) for a diffusion-reaction equation in a bulk domain which is coupled to a corresponding equation on the boundary of the bulk domain. The bulk domain is embedded into a structured, unfitted Background Mesh. By adding certain stabilization terms to the discrete variational formulation of the coupled bulk-surface problem, the resulting cutDGM is provably stable and exhibits optimal convergence properties as demonstrated by numerical experiments. We also show both theoretically and numerically that the system matrix is well-conditioned, irrespective of the relative position of the bulk domain in the Background Mesh.
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Cut Finite Element Methods for Partial Differential Equations on Embedded Manifolds of Arbitrary Codimensions
arXiv: Numerical Analysis, 2016Co-Authors: Erik Burman, Peter Hansbo, Mats G. Larson, André MassingAbstract:We develop a theoretical framework for the analysis of stabilized cut finite element methods for the Laplace-Beltrami operator on a manifold embedded in $\mathbb{R}^d$ of arbitrary codimension. The method is based on using continuous piecewise polynomials on a Background Mesh in the embedding space for approximation together with a stabilizing form that ensures that the resulting problem is stable. The discrete manifold is represented using a triangulation which does not match the Background Mesh and does not need to be shape-regular, which includes level set descriptions of codimension one manifolds and the non-matching embedding of independently triangulated manifolds as special cases. We identify abstract key assumptions on the stabilizing form which allow us to prove a bound on the condition number of the stiffness matrix and optimal order a priori estimates. The key assumptions are verified for three different realizations of the stabilizing form including a novel stabilization approach based on penalizing the surface normal gradient on the Background Mesh. Finally, we present numerical results illustrating our results for a curve and a surface embedded in $\mathbb{R}^3$.
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A Cut Discontinuous Galerkin Method for the Laplace-Beltrami Operator
arXiv: Numerical Analysis, 2015Co-Authors: Erik Burman, Peter Hansbo, Mats G. Larson, André MassingAbstract:We develop a discontinuous cut finite element method (CutFEM) for the Laplace-Beltrami operator on a hypersurface embedded in $\mathbb{R}^d$. The method is constructed by using a discontinuous piecewise linear finite element space defined on a Background Mesh in $\mathbb{R}^d$. The surface is approximated by a continuous piecewise linear surface that cuts through the Background Mesh in an arbitrary fashion. Then a discontinuous Galerkin method is formulated on the discrete surface and in order to obtain coercivity, certain stabilization terms are added on the faces between neighboring elements that provide control of the discontinuity as well as the jump in the gradient. We derive optimal a priori error and condition number estimates which are independent of the positioning of the surface in the Background Mesh. Finally, we present numerical examples confirming our theoretical results.
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A Stabilized Cut Finite Element Method for the Three Field Stokes Problem
SIAM Journal on Scientific Computing, 2015Co-Authors: Erik Burman, Susanne Claus, André MassingAbstract:We propose a Nitsche-based fictitious domain method for the three field Stokes problem in which the boundary of the domain is allowed to cross through the elements of a fixed Background Mesh. The dependent variables of velocity, pressure, and extra-stress tensor are discretized on the Background Mesh using linear finite elements. This equal order approximation is stabilized using a continuous interior penalty (CIP) method. On the unfitted domain boundary, Dirichlet boundary conditions are weakly enforced using Nitsche's method. We add CIP-like ghost penalties in the boundary region and prove that our scheme is inf-sup stable and that it has optimal convergence properties independent of how the domain boundary intersects the Mesh. Additionally, we demonstrate that the condition number of the system matrix is bounded independently of the boundary location. We corroborate our theoretical findings numerically.
Wim Desmet - One of the best experts on this subject based on the ideXlab platform.
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Stability-preserving model order reduction for time-domain simulation of vibro-acoustic FE models
International Journal for Numerical Methods in Engineering, 2017Co-Authors: A. Van de Walle, Elke Deckers, Frank Naets, Wim DesmetAbstract:We describe a method for discretizing planar C2-regular domains immersed in non-conforming triangulations. The method consists in constructing mappings from triangles in a Background Mesh to curvilinear ones that conform exactly to the immersed domain. Constructing such a map relies on a novel way of parameterizing the immersed boundary over a collection of nearby edges with its closest point projection. By interpolating the mappings to curvilinear triangles at select points, we recover isoparametric mappings for the immersed domain defined over the Background Mesh. Indeed, interpolating the constructed mappings just at the vertices of the Background Mesh yields a fast Meshing algorithm that involves only perturbing a few vertices near the boundary. For the discretization of a curved domain to be robust, we have to impose restrictions on the Background Mesh. Conversely, these restrictions define a family of domains that can be discretized with a given Background Mesh. We then say that the Background Mesh is a universal Mesh for such a family of domains. The notion of universal Meshes is particularly useful in free/moving boundary problems because the same Background Mesh can serve as the universal Mesh for the evolving domain for time intervals that are independent of the time step. Hence it facilitates a framework for finite element calculations over evolving domains while using a fixed Background Mesh. Furthermore, since the evolving geometry can be approximated with any desired order, numerical solutions can be computed with high-order accuracy. We demonstrate these ideas with various numerical examples.