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Parametric FEM for Geometric Biomembranes
, 2011
"... We consider geometric biomembranes governed by an L²gradient flow for bending energy subject to area and volume constraints (Helfrich model). We give a concise derivation of a novel vector formulation, based on shape differential calculus, and corresponding discretization via parametric FEM using q ..."
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We consider geometric biomembranes governed by an L²gradient flow for bending energy subject to area and volume constraints (Helfrich model). We give a concise derivation of a novel vector formulation, based on shape differential calculus, and corresponding discretization via parametric FEM using quadratic isoparametric elements and a semiimplicit Euler method. We document the performance of the new parametric FEM with a number of simulations leading to dumbbell, red blood cell and toroidal equilibrium shapes while exhibiting large deformations.
A Hybrid Variational Front TrackingLevel Set Mesh Generator For Problems Exhibiting Large Deformations and Topological Changes
"... We present a method for generating 2D unstructured triangular meshes that undergo large deformations and topological changes in an automatic way. We employ a method for detecting when topological changes are imminent via distance functions and shape skeletons. When a change occurs, we use a level s ..."
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We present a method for generating 2D unstructured triangular meshes that undergo large deformations and topological changes in an automatic way. We employ a method for detecting when topological changes are imminent via distance functions and shape skeletons. When a change occurs, we use a level set method to guide the change of topology of the domain mesh. This is followed by an optimization procedure, using a variational formulation of active contours, that seeks to improve boundary mesh conformity to the zero level contour of the level set function. Our method is advantageous for ArbitraryLagrangianEulerian (ALE) type methods and directly allows for using a variational formulation of the physics being modeled and simulated, including the ability to account for important geometric information in the model (such as for surface tension driven flow). Furthermore, the meshing procedure is not required at every timestep and the level set update is only needed during a topological change. Hence, our method does not significantly affect computational cost. Key words:
Optimal Shape Design for the Viscous Incompressible Flow ∗
, 2008
"... Abstract. This paper is concerned with a numerical simulation of shape optimization in a twodimensional viscous incompressible flow governed by Navier–Stokes equations with mixed boundary conditions containing the pressure. The minimization problem of total dissipated energy was established in the ..."
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Abstract. This paper is concerned with a numerical simulation of shape optimization in a twodimensional viscous incompressible flow governed by Navier–Stokes equations with mixed boundary conditions containing the pressure. The minimization problem of total dissipated energy was established in the fluid domain. We derive the structures of shape gradient of the cost functional by using the differentiability of a minimax formulation involving a Lagrange functional with a function space parametrization technique. Finally a gradient type algorithm is effectively used for our problem. Keywords. shape optimization; minimax principle; gradient algorithm; Navier– Stokes equations.
A VARIATIONAL SHAPE OPTIMIZATION APPROACH FOR IMAGE SEGMENTATION WITH A MUMFORDSHAH FUNCTIONAL ∗
"... Abstract. We introduce a novel computational method for a MumfordShah functional, which decomposes a given image into smooth regions separated by closed curves. Casting this as a shape optimization problem, we develop a gradient descent approach at the continuous level that yields nonlinear PDE flo ..."
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Abstract. We introduce a novel computational method for a MumfordShah functional, which decomposes a given image into smooth regions separated by closed curves. Casting this as a shape optimization problem, we develop a gradient descent approach at the continuous level that yields nonlinear PDE flows. We propose time discretizations that linearize the problem, and space discretization by continuous piecewise linear finite elements. The method incorporates topological changes, such as splitting and merging for detection of multiple objects, spacetime adaptivity and a coarsetofine approach to process large images efficiently. We present several simulations that illustrate the performance of the method, and investigate the model sensitivity to various parameters. Key words. image segmentation, MumfordShah, shape optimization, finite element method AMS subject classifications. 49M15,49M25,65D15,65K10,68T45,90C99
FIRST VARIATION OF THE GENERAL CURVATUREDEPENDENT SURFACE ENERGY
"... Abstract. We consider general surface energies, which are weighted integrals over a closed surface with a weight function depending on the position, the unit normal and the total curvature of the surface. Energies of this form have applications in many areas, such as materials science, biology and i ..."
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Abstract. We consider general surface energies, which are weighted integrals over a closed surface with a weight function depending on the position, the unit normal and the total curvature of the surface. Energies of this form have applications in many areas, such as materials science, biology and image processing. Often one is interested in finding a surface that minimizes such an energy, which entails finding its first variation with respect to perturbations of the surface. We present a concise derivation of the first variation of the general surface energy using tools from shape differential calculus. We first derive a scalar strong form and next a vector weak form of the first variation. The latter reveals the variational structure of the first variation, avoids dealing explicitly with the second fundamental form, and thus can be easily discretized using parametric finite elements. Our results are valid for surfaces in any number of dimensions and unify all previous results derived for specific examples of such surface energies.
AFEM FOR SHAPE OPTIMIZATION
"... Abstract. We examine shape optimization problems in the context of inexact sequential quadratic programming. Inexactness is a consequence of using adaptive finite element methods (AFEM) to approximate the state and adjoint equations (via the dual weighted residual method), update the boundary, and c ..."
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Abstract. We examine shape optimization problems in the context of inexact sequential quadratic programming. Inexactness is a consequence of using adaptive finite element methods (AFEM) to approximate the state and adjoint equations (via the dual weighted residual method), update the boundary, and compute the geometric functional. We present a novel algorithm that equidistributes the errors due to shape optimization and discretization, thereby leading to coarse resolution in the early stages and fine resolution upon convergence, and thus optimizing the computational effort. We discuss the ability of the algorithm to detect whether or not geometric singularities such as corners are genuine to the problem or simply due to lack of resolution—a new paradigm in adaptivity. 1. Shape Optimization as Adaptive Sequential Quadratic Programming We consider shape optimization problems for partial differential equations (PDE) that can be formulated as follows: We denote with u = u(Ω) the solution of a PDE in the domain Ω of R d (d ≥ 2), (1) Bu(Ω) = f, which we call the state equation. Given a cost functional J[Ω] = J[Ω,u(Ω)], which depends on Ω itself and the solution u(Ω) of the state equation, we consider the minimization problem (2) Ω ∗ ∈ Uad: J[Ω ∗,u(Ω ∗)] = inf J[Ω,u(Ω)], Ω∈Uad where Uad is a set of admissible domains in Rd. We view this as a constrained minimization problem, (1) being the constraint. We do not discuss conditions on B,J or Uad that yield existence of a solution. The goal of this paper is, instead, to formulate and test a practical and efficient computational algorithm that adaptively builds a sequence of domains {Ωk}k≥0 converging to a local minimizer Ω of (1)–(2). Coupling adaptivity with shape optimization seems to be important but rather novel. To achieve this goal we will define an Adaptive Sequential Quadratic Programming algorithm (ASQP). To motivate and briefly describe the ideas underlying ASQP, we need the concept of shape derivative δΩJ[Ω;v] of J[Ω] in the direction of a velocity v, which usually satisfies (3) δΩJ[Ω;v] = g(Ω)v dS = 〈g(Ω),v〉Γ,