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The Explicit Jump Immersed Interface Method and Interface Problems for Differential Equations
, 1998
"... The Explicit Jump Immersed Interface Method and Interface Problems for Differential Equations by Andreas Wiegmann Chairperson of Supervisory Committee: Professor Kenneth P. Bube Department of Mathematics We study and numerically solve elliptic differential equations in the presence of interfaces wh ..."
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Cited by 51 (7 self)
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The Explicit Jump Immersed Interface Method and Interface Problems for Differential Equations by Andreas Wiegmann Chairperson of Supervisory Committee: Professor Kenneth P. Bube Department of Mathematics We study and numerically solve elliptic differential equations in the presence of interfaces where the solution is not smooth. We use uniform Cartesian grids and do not require the interfaces to be aligned with the grid. We develop a onedimensional theory for the new Explicit Jump Immersed Interface Method (EJIIM), which culminates in a proof of secondorder convergence for piecewiseconstant coefficients for singlepoint interfaces. The proof is interesting in not requiring the numerical scheme to satisfy a discrete maximum principle, the usual means by which such results are proved, and in providing error bounds that are independent of the geometry and the contrast in the coefficients. EJIIM works by focusing on the jumps in the solutions and their derivatives, rather than on findin...
A Framework for the Construction of Level Set Methods for Shape Optimization and Reconstruction
 Interfaces and Free Boundaries
, 2002
"... The aim of this paper is to develop a functionalanalytic framework for the construction of level set methods, when applied to shape optimization and shape reconstruction problems. As a main tool we use a notion of gradient ows for geometric configurations such as used in the modelling of geometric ..."
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Cited by 46 (6 self)
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The aim of this paper is to develop a functionalanalytic framework for the construction of level set methods, when applied to shape optimization and shape reconstruction problems. As a main tool we use a notion of gradient ows for geometric configurations such as used in the modelling of geometric motions in materials science. The analogies to this field lead to a scale of level set evolutions, characterized by the norm used for the choice of the velocity. This scale of methods also includes the standard approach used in previous work on this subject as a special case.
An immersed interface method for viscous incompressible flows involving rigid and flexible boundaries
 J. Comp. Phys
, 2006
"... We present an immersed interface method for the incompressible NavierStokes equations capable of handling rigid immersed boundaries. The immersed boundary is represented by a set of Lagrangian control points. In order to guarantee that the noslip condition on the boundary is satisfied, singular fo ..."
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Cited by 37 (3 self)
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We present an immersed interface method for the incompressible NavierStokes equations capable of handling rigid immersed boundaries. The immersed boundary is represented by a set of Lagrangian control points. In order to guarantee that the noslip condition on the boundary is satisfied, singular forces are applied on the fluid. The forces are related to the jumps in pressure and the jumps in the derivatives of both pressure and velocity, and are interpolated using cubic splines. The strength of the singular forces is determined by solving a small system of equations iteratively at each time step. The NavierStokes equations are discretized on a staggered Cartesian grid by a second order accurate projection method for pressure and velocity. Keywords: Immersed interface method, NavierStokes equations, Cartesian grid method, finite difference, fast Poisson solvers, irregular domains.
Topology optimization with implicit functions and regularization
 INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING 2003
, 2003
"... Topology optimization is formulated in terms of the nodal variables that control an implicit function description of the shape. The implicit function is constrained by upper and lower bounds, so that only a band of nodal variables needs to be considered in each step of the optimization. The weak for ..."
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Cited by 36 (2 self)
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Topology optimization is formulated in terms of the nodal variables that control an implicit function description of the shape. The implicit function is constrained by upper and lower bounds, so that only a band of nodal variables needs to be considered in each step of the optimization. The weak form of the equilibrium equation is expressed as a Heaviside function of the implicit function; the Heaviside function is regularized to permit the evaluation of sensitivities. We show that the method is a dual of the Bendsøe–Kikuchi method. The method is applied both to problems of optimizing single material and multimaterial configurations; the latter is made possible by enrichment functions based on the extended finite element method that enable discontinuous derivatives to be accurately treated within an element. The method is remarkably robust and we found no instances of checkerboarding. The method handles topological merging and separation without any apparent difficulties.
Composite Finite Elements for 3D Image Based Computing
 COMPUTING AND VISUALIZATION IN SCIENCE 12
, 2009
"... We present an algorithmical concept for modeling and simulation with partial differential equations (PDEs) in image based computing where the computational geometry is defined through previously segmented image data. Such problems occur in applications from biology and medicine where the underlying ..."
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Cited by 28 (4 self)
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We present an algorithmical concept for modeling and simulation with partial differential equations (PDEs) in image based computing where the computational geometry is defined through previously segmented image data. Such problems occur in applications from biology and medicine where the underlying image data has been acquired through, e.g. computed tomography (CT), magnetic resonance imaging (MRI) or electron microscopy (EM). Based on a levelset description of the computational domain, our approach is capable of automatically providing suitable composite finite element functions that resolve the complicated shapes in the medical/biological data set. It is efficient in the sense that the traversal of the grid (and thus assembling matrices for finite element computations) inherits the efficiency of uniform grids away from complicated structures. The method’s efficiency heavily depends on precomputed lookup tables in the vicinity of the domain boundary or interface. A suitable multigrid method is used for an efficient solution of the systems of equations resulting from the composite finite element discretization. The paper focuses on both algorithmical and implementational details. Scalar and vector valued model problems as well as real applications underline the usability of our approach.
A levelset based variational method for design and optimization of heterogeneous objects
, 2004
"... A heterogeneous object is referred to as a solid object made of different constituent materials. The object is of a finite collection of regions of a set of prescribed material classes of continuously varying material properties. These properties have a discontinuous change across the interface of t ..."
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Cited by 26 (6 self)
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A heterogeneous object is referred to as a solid object made of different constituent materials. The object is of a finite collection of regions of a set of prescribed material classes of continuously varying material properties. These properties have a discontinuous change across the interface of the material regions. In this paper, we propose a levelset based variational approach for the design of this class of heterogeneous objects. Central to the approach is a variational framework for a wellposed formulation of the design problem. In particular, we adapt the MumfordShah model which specifies that any point of the object belongs to either of two types: inside a material region of a welldefined gradient or on the boundary edges and surfaces of discontinuities. Furthermore, the set of discontinuities is represented implicitly, using a multiphase level set model. This levelset based variational approach yields a computational system of coupled geometric evolution and diffusion partial differential equations. Promising features of the proposed method include strong regularity in the problem formulation and inherent capabilities of geometric and material modeling, yielding a common framework for optimization of the heterogeneous objects that incorporates dimension, shape, topology, and material properties. The proposed method is illustrated with several 2D examples of optimal design of multimaterial structures and materials.
A LevelSet Method for Vibration and Multiple Loads Structural Optimization
, 2004
"... We extend the levelset method for shape and topology optimization to new objective functions such as eigenfrequencies and multiple loads. This method ..."
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Cited by 25 (3 self)
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We extend the levelset method for shape and topology optimization to new objective functions such as eigenfrequencies and multiple loads. This method
Structural shape and topology optimization in a level set . . .
 SUBMITTED TO STRUCTURAL AND MULTIDISCIPLINARY OPTIMIZATION
, 2002
"... In this paper we present a new framework to approach the problem of structural shape and topology optimization. We use a level set method with an implicit moving boundary model. As a boundary optimization problem, the structural boundary description is implicitly embedded in a scalar function as its ..."
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Cited by 18 (5 self)
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In this paper we present a new framework to approach the problem of structural shape and topology optimization. We use a level set method with an implicit moving boundary model. As a boundary optimization problem, the structural boundary description is implicitly embedded in a scalar function as its “isosurfaces. ” Such level set models are flexible in handling complex topological changes and are concise in describing the boundary shape of the structure. Furthermore, by using a simple HamiltonJacobi convection equation, the movement of the implicit moving boundaries of the structure is driven by a transformation of the objective and the constraints into a speed function that defines the level set propagation. The result is a 3D structural optimization technique that demonstrates outstanding flexibility of handling topological changes, fidelity of boundary representation and degree of automation, comparing favorably with other methods based on explicit boundary variation or homogenization in the literature. We have developed a number of numerical techniques for an efficient and robust implementation of the proposed method. The method is tested with several examples of a linear elastic structure that are widely reported in the topology optimization literature.
Design of multimaterial compliant mechanisms using level set methods
 ASME Transaction: Journal of Mechanical Design
, 2005
"... A monolithic compliant mechanism transmits applied forces from specified input ports to output ports by elastic deformation of its comprising materials, fulfilling required functions analogous to a rigidbody mechanism. In this paper, we propose a levelset method for designing monolithic compliant ..."
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Cited by 16 (2 self)
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A monolithic compliant mechanism transmits applied forces from specified input ports to output ports by elastic deformation of its comprising materials, fulfilling required functions analogous to a rigidbody mechanism. In this paper, we propose a levelset method for designing monolithic compliant mechanisms made of multiple materials as an optimization of continuum heterogeneous structures. Central to the method is a multiphase level set model that precisely specifies the distinct material regions and their sharp interfaces as well as the geometric boundary of the structure. Combined with the classical shape derivatives, the levelset method yields an Eulerian computational system of geometric partial differential equations, capable of performing topological changes and capturing geometric evolutions at the interface and the boundary. The proposed method is demonstrated for singleinput and singleoutput mechanisms and illustrated with several 2D examples of synthesis of multimaterial mechanisms of forceinverters and gripping and clamping devices. An analysis on the formation of de facto hinges is presented based on the shape gradient information. A scheme to ensure a wellconnected topology of the mechanism during the process of optimization is also presented.