A New Level-Set Model for the Representation of Non-Smooth Geometries

In Starinshak et al. (J Comput Phys 262(1):1–16, 2014), we proposed a new level-set model for representing multimaterial flows in multiple space dimensions. Rather than associating each level-set function with the boundary of a material, the new model associates each level-set function with a pair of materials and the interface that separates them. In this paper, we extend the model to represent geometries with non-smooth boundaries. The model uses multiple level-set functions to describe the shape boundary, typically with one level-set function per smooth boundary segment. Sign information is collected from all level-set functions and a voting algorithm is used to determine the interior/exterior of the geometric shape. The model is well suited for representing boundaries with singularities; it offers significant improvement over standard level-set approaches, both in shape preservation and area conservation; and it eliminates the need for costly redistancing of the level-set function. Numerical examples illustrate the superior performance of the proposed model.     

Semi-Lagrange method for level-set-based structural topology and shape optimization

In this paper, we introduce a semi-Lagrange scheme to solve the level-set equation in structural topology optimization. The level-set formulation of the problem expresses the optimization process as a solution to a Hamilton–Jacobi partial differential equation. It allows for the use of shape sensitivity to derive a speed function for a descent solution. However, numerical stability condition in the explicit upwind scheme for discrete level-set equation severely restricts the time step, requiring a large number of time steps for a numerical solution. To improve the numerical efficiency, we propose to employ a semi-Lagrange scheme to solve level-set equation. Therefore, a much larger time step can be obtained and a much smaller number of time steps are required. Numerical experiments comparing the semi-Lagrange method with the classical explicit upwind scheme are presented for the problem of mean compliance optimization in two dimensions.     

Introducing the sequential linear programming level-set method for topology optimization

This paper introduces an approach to level-set topology optimization that can handle multiple constraints and simultaneously optimize non-level-set design variables. The key features of the new method are discretized boundary integrals to estimate function changes and the formulation of an optimization sub-problem to attain the velocity function. The sub-problem is solved using sequential linear programming (SLP) and the new method is called the SLP level-set method. The new approach is developed in the context of the Hamilton-Jacobi type level-set method, where shape derivatives are employed to optimize a structure represented by an implicit level-set function. This approach is sometimes referred to as the conventional level-set method. The SLP level-set method is demonstrated via a range of problems that include volume, compliance, eigenvalue and displacement constraints and simultaneous optimization of non-level-set design variables.     

Shape equilibrium constraint: a strategy for stress-constrained structural topology optimization

In topology optimization of a continuum, it is important to consider stress-related objective or constraints, from both theoretical and application perspectives. It is known that the problem is challenging. Although remarkable achievements have been made with the SIMP (Solid Isotropic Material with Penalization) framework, a number of critical issues are yet to be fully resolved. In the paper, we present an approach of a shape equilibrium constraint strategy with the level-set/X-FEM framework. We formulate the topology optimization problem under (spatially-distributed) stress constraints into a shape equilibrium problem of active stress constraint. This formulation allows us to effectively handle the stress constraint, and the intrinsic non-differentiability introduced by local stress constraints is removed. The optimization problem is made into one of continuous shape-sensitivity and it is solved by evolving a coherent interface of the shape equilibrium concurrently with shape variation in the structural boundary during a level-set evolution process. Several numerical examples in two dimensions are provided as a benchmark test of the proposed shape equilibrium constraint strategy for minimum-weight and fully-stressed designs and for designs with stress constraint satisfaction.     

A variational binary level-set method for elliptic shape optimization problems

We present a variational binary level-set method to solve a class of elliptic problems in shape optimization. By the ‘ersatz material’ approach, which amounts to fill the holes by a weak phase, the original shape optimization model is approximated by a two-phase optimization problem. Under the binary level-set framework, we need to optimize a smooth functional under a binary constraint. We propose an augmented Lagrangian method to solve the constrained optimization problem. Numerical results are presented and compared with those obtained by level-set methods, which demonstrate the robustness and efficiency of our method.     

Optimal distribution of multimaterial composites for torsional beams

In this paper, we consider the optimal design of torsional beams using an arbitrary number of materials. The problem is initially ill-posed, and must be relaxed by the introduction of multimaterial composites. The optimization algorithm for multimaterial composites is described and computational results for both perturbations and asymptotical cases are presented. It is noticed that the case for three or more composites undergoes a phenomenon very much like a phase transition when certain conditions are satisfied.     

Wear of Polysilicon Surface Micromachines Operated in High Vacuum

The evolution of wear at sidewall surfaces of polysilicon microelectromechanical systems was investigated in high vacuum under controlled normal load and sliding speed conditions. The static adhesion force was used as an indicator of the changes in wear characteristics occurring during oscillatory sliding contact. Measurements of the static adhesion force as a function of sliding cycles and scanning electron microscopy observations of micromachines from the same batch process subjected to nominally identical testing conditions revealed two distinctly different tribological patterns, namely, low-adhesion/high-wear behavior and high-adhesion/low-wear behavior. The static adhesion force and wear behavior were found to be in direct correlation with the micromachine operational lifetime. Transmission electron microscopy, selected area diffraction, and energy dispersive X-ray spectroscopy yielded insight into the origin, microstructure, and composition of wear debris and agglomerates adhered onto the sliding surfaces. Results demonstrate a strong dependence of micromachine operational life on the removal of the native oxide film and the organic monolayer coating as well as the formation of agglomerates consisting of organic coating material and wear debris.     

A level-set based variational method for design and optimization of heterogeneous objects

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 level-set based variational approach for the design of this class of heterogeneous objects. Central to the approach is a variational framework for a well-posed formulation of the design problem. In particular, we adapt the Mumford-Shah model which specifies that any point of the object belongs to either of two types: inside a material region of a well-defined gradient or on the boundary edges and surfaces of discontinuities. Furthermore, the set of discontinuities is represented implicitly, using a multi-phase level set model. This level-set 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 multi-material structures and materials.     

Design of thermoelastic multi-material structures with graded interfaces using topology optimization

A level-set based shape and topology optimization framework is used to investigate the influence of graded interfaces in the optimization of micro-architectured multi-materials. In contrast to other studies found in the literature, interfaces are considered as smooth and graded transitions between constitutive phases instead of sharp delimitations. Case studies for extreme thermoelastic properties of 2D isotropic composites are analyzed and optimal designs are presented. It is shown that explicitly accounting for interfaces can influence the design of heterogeneous materials in composite microstructures.     

A level-set method for vibration and multiple loads structural optimization

We extend the level-set method for shape and topology optimization to new objective functions such as eigenfrequencies and multiple loads. This method is based on a combination of the classical shape derivative and of the Osher–Sethian level-set algorithm for front propagation. In two and three space dimensions we maximize the first eigenfrequency or we minimize a weighted sum of compliances associated to different loading configurations. The shape derivative is used as an advection velocity in a Hamilton–Jacobi equation for changing the shape. This level-set method is a low-cost shape capturing algorithm working on a fixed Eulerian mesh and it can easily handle topology changes.     

Fabrication, metallographic and tribologic investigations of LIGA-microstructures made of nickelphosphorous alloys

 The performance and lifetime of micro electro mechanical systems (MEMS) is strongly affected by friction and wear. Because of the restrictions due to the fabrication process the variety of materials used for micro systems is not very manifold. This results often is very poor tribological properties, since the tribological pairings are disadvantageous with regard to friction and especially wear. We therefore investigated materials which can be fabricated by the process currently used for LIGA-microstructures, and have the potential of better properties concerning friction and wear. The results of our tribologic experiments showed, that nickel phosphorous alloys are a promising material for microsystems suffering wear, especially at high surface pressures (high loading). Their absolute values of the wear intensity, are at least one order of magnitude lower than those of nickel and copper, which are the materials mostly used today. Received: 27 June 1996/Accepted:12 July 1996     

On the Discrete-Time Integral Sliding-Mode Control

A new discrete-time integral sliding-mode control (DISMC) scheme is proposed for sampled-data systems. The new control scheme is characterized by a discrete-time integral sliding manifold which inherits the desired properties of the continuous-time integral sliding manifold, such as full order sliding manifold with pole assignment, and elimination of the reaching phase. In particular, comparing with existing discrete-time sliding-mode control, the new scheme is able to achieve more precise tracking performance. It will be shown in this work that, the new control scheme achieves O(T²) steady-state error for state regulation with the widely adopted delay-based disturbance estimation. Another desirable feature is, the proposed DISMC prevents the generation of overlarge control actions due to deadbeat response, which is usually inevitable due to the existence of poles at the origin for a reduced order sliding manifold designed for sampled-data systems. Both the theoretical analysis and illustrative example demonstrate the validity of the proposed scheme     

Multi-constrained topology optimization via the topological sensitivity

The objective of this paper is to introduce and demonstrate a robust method for multi-constrained topology optimization. The method is derived by combining the topological sensitivity with the classic augmented Lagrangian formulation.The primary advantages of the proposed method are: (1) it rests on well-established augmented Lagrangian formulation for constrained optimization, (2) the augmented topological level-set can be derived systematically for an arbitrary set of loads and constraints, and (3) the level-set can be updated efficiently. The method is illustrated through numerical experiments.     

Topology optimization of shell-infill structures using a distance regularized parametric level-set method

This paper presents a novel topology optimization formulation for shell-infill structures based on a distance regularized parametric level-set method (PLSM). In this method, the outer shell and the infill are represented by two distinct level sets of a single-level set function (LSF). In order to obtain a controllable and uniform shell thickness, a distance regularization (DR) term is introduced to formulate a weighted bi-objective function. The DR term is minimized along with the original objective, regularizing the parametric LSF close to a signed distance function. With the signed distance property, the area between the two-level sets can be contoured as the shell with a uniform thickness. Additionally, the presented formulation retains one important merit of the PLSM that new holes are able to nucleate during the optimization process. With respect to the material of the shell, the infill is filled with a weaker and lighter material with tunable parameters. Particularly, the infill can be pre-designed with isotropic microstructures. Three compliance minimization examples are provided to demonstrate the effectiveness of this formulation.

     

Topology optimization under thermo-elastic buckling

The focus of this paper is on topology optimization of continuum structures subject to thermally induced buckling. Popular strategies for solving such problems include Solid Isotropic Material with Penalization (SIMP) and Rational Approximation of Material Properties (RAMP). Both methods rely on material parameterization, and can sometimes exhibit pseudo buckling modes in regions with low pseudo-densities. Here we consider a level-set approach that relies on the concept of topological sensitivity. Topological sensitivity analysis for thermo-elastic buckling is carried out via direct and adjoint formulations. Then, an augmented Lagrangian formulation is presented that exploits these sensitivities to solve a buckling constrained problem. Numerical experiments in 3D illustrate the robustness and efficiency of the proposed method.     

Experimental application and enhancement of the XFEM–GA algorithm for the detection of flaws in structures

The extended finite element formulation (XFEM) combined with genetic algorithms (GAs) have previously been shown to be very effective in the detection of flaws in structures. By this approach, the XFEM is used to model the forward problem and a GA is used as the optimization scheme, converging to the true flaw. The convergence is obtained by minimizing the error between sensor measurements and data obtained by solving the forward problem.The current study proposes several advances of this XFEM–GA algorithm, more specifically: (i) a novel genetic algorithm that accelerates the convergence of the scheme and alleviates entrapment in local optima, (ii) a generic XFEM formulation of an elliptical hole which is utilized to detect any type of flaw (cracks or holes) of any shape, and (iii) experimental verification of the approach for an arbitrary crack in a 2D plate.Convergence studies on various benchmark problems including the experimental verification clearly show the potential of this approach to detection of arbitrary flaws.     

Tensor-SIFT Based Earth Mover’s Distance for Contour Tracking

Contour tracking in complex environments is a difficult problem due to the cluttered backgrounds, illumination changes, occlusion and camera viewpoint variations etc. This paper presents a region functional based on the Earth Mover’s Distance (EMD), computation of which is mathematically modeled as the transportation problem (TP), for robust contour tracking in the challenging conditions. Formulation of EMD-based functional can be described as variational EMD (VEMD) since the contour curve function is involved for optimization. Minimizing the EMD-based functional is nontrivial and we develop a two-phase method for its optimization. In the first phase, letting the candidate contour be fixed, we seek the best solution to the TP by the Simplex algorithm. Then through the shape derivative theory, we make a perturbation analysis of the contour around the best solution to the TP. As a result we obtain a partial differential equation (PDE) that is solved by the level-set algorithm. The two-phase procedure iterates until the appropriate stopping criterions are satisfied. Alongside the EMD-based functional formulation, we introduce a dimensionality reduction method by tensor decomposition, achieving a low-dimensional Tensor-SIFT features for object representation. Applicable to both the color and gray-level images, Tensor-SIFT features are distinctive, insensitive to illumination and viewpoint changes. Finally, we develop an integrated algorithm that combines various techniques, e.g. the Simplex algorithm, narrow-band level set and fast marching algorithms. Particularly, we provide a method for the level-set initialization between two successive frames and the criterions for stopping the iterative functional optimization. Experiments in challenging image sequences show that the proposed work has promising performance.