A Topology-Preserving Level Set Method for Shape Optimization

Topology design optimization offers tremendous opportunity in design and manufacturing freedoms by designing and producing a part from the ground-up without a meaningful initial design as required by conventional shape design optimization approaches. Ideally, with adequate problem statements, to formulate and solve the topology design problem using a standard topology optimization process, such as SIMP (Simplified Isotropic Material with Penalization) is possible. In reality,...

In this paper, we propose a level set-based shape optimization method for acoustic wave propagation problems with a deformable structure. First, we propose a mathematical model for acoustic wave propagation with a deformed structure based on coordinate transformation and the Eulerian approach. Next, we formulate the shape optimization problem and perform sensitivity analysis based on the shape derivative concept. We then construct an optimization algorithm based on the framew...

We exploit level set topology optimization to find the optimal material distribution for metamaterial-based heat manipulators. The level set function, geometry, and solution field are parameterized using the non-uniform rational B-spline (NURBS) basis functions in order to take advantage of easy control of smoothness and continuity. In addition, NURBS approximations can produce conic geometries exactly and provide higher efficiency for higher-order elements. The values of the...

We introduce a unified sensitivity concept for shape and topological perturbations and perform the sensitivity analysis for a discretized PDE-constrained design optimization problem in two space dimensions. We assume that the design is represented by a piecewise linear and globally continuous level set function on a fixed finite element mesh and relate perturbations of the level set function to perturbations of the shape or topology of the corresponding design. We illustrate ...

We propose here a proof of existence of a minimizer of a segmentation functional based on a priori information on target shapes, and formulated with level sets. The existence of a minimizer is very important, because it guarantees the convergence of any numerical methods (either gradient descents techniques and variants, or PDE resolutions) used to solve the segmentation model. This work can also be used in many other segmentation models to prove the existence of a minimizer.

We propose a geometric convexity shape prior preservation method for variational level set based image segmentation methods. Our method is built upon the fact that the level set of a convex signed distanced function must be convex. This property enables us to transfer a complicated geometrical convexity prior into a simple inequality constraint on the function. An active set based Gauss-Seidel iteration is used to handle this constrained minimization problem to get an efficie...

In this paper, we study triply periodic surfaces with minimal surface area under a constraint in the volume fraction of the regions (phases) that the surface separates. Using a variational level set method formulation, we present a theoretical characterization of and a numerical algorithm for computing these surfaces. We use our theoretical and computational formulation to study the optimality of the Schwartz P, Schwartz D, and Schoen G surfaces when the volume fractions of t...

This paper presents an efficient 51 lines Matlab code to solve topology optimization problems. By the fact that the presented code is based on an hard 0-1 optimization method that handles the integer part of the optimization in a simple fashion and in sub-polynomial time, it has been possible to not only reduce the number of lines to 51 without sacrificing any readability, but also to obtain void-solid designs with close to none checkerboard patterns with improved efficiency....

Convex optimization problems arising in applications often have favorable objective functions and complicated constraints, thereby precluding first-order methods from being immediately applicable. We describe an approach that exchanges the roles of the objective and constraint functions, and instead approximately solves a sequence of parametric level-set problems. A zero-finding procedure, based on inexact function evaluations and possibly inexact derivative information, lead...

High-order (non-linear) functionals have become very popular in segmentation, stereo and other computer vision problems. Level sets is a well established general gradient descent framework, which is directly applicable to optimization of such functionals and widely used in practice. Recently, another general optimization approach based on trust region methodology was proposed for regional non-linear functionals. Our goal is a comprehensive experimental comparison of these two...