Level Sets Of Value Function Research Articles

A local solution approach for level-set based structural topology optimization in isogeometric analysis

Abstract This paper develops a new topology optimization approach for minimal compliance problems based on the parameterized level set method in isogeometric analysis. Here, we choose the basis functions as level set functions. The design variables are obtained with Greville abscissae based on the corresponding collocation points. The zero-level set boundaries are derived from the level set function values of the interpolation points in all knot spans. In the optimization iteration process, the whole design domain is discretized into two types of subdomains around the zero-level set boundaries, undesign area with void materials and redesign domain with solid materials. To decrease the size of equations and the computational consumptions, only the solid material area is recalculated and the void material area is discarded according to the high accuracy of isogeometric analysis. Numerical examples demonstrate the validity of the proposed optimization method.

A stable generalized finite element method (SGFEM) of degree two for interface problems

Generalized or Extended Finite Element Methods (GFEM/XFEM) of degree 1 for interface problems have been reported in the literature; they (i) yield optimal order of convergence, i.e., O(h) (h is a discretization parameter), (ii) are stable in a sense that conditioning is not worse than that of the standard FEM, and (iii) are robust in that the conditioning does not deteriorate as interface curves are close to boundaries of underlying elements. However, higher order GFEM/XFEM with the properties (i)–(iii) have not been successfully addressed yet. Various enrichment schemes for GFEM/XFEM based on D or DPk (D is a distance function or the absolute value of level set function, and Pk is the polynomial basis of degree k) have been reported recently to obtain higher order convergence, but they are not stable or robust in general; in fact, they even may not yield the optimal orders of convergence. In this paper, we propose a stable GFEM/XFEM of degree 2 (SGFEM2) for the interface problems, where we used the enrichment scheme based on D{1,x,y}, instead of D or D{1,x,y,x2,xy,y2} in the literature. We prove that the SGFEM2 yields the optimal order of convergence, i.e., O(h2), for the interface problems with curved (smooth) interfaces. A local principal component analysis technique has been proposed, which ensures that the SGFEM2 is stable and robust. Numerical experiments for straight and curved interfaces have been presented to illuminate these properties.

A robust sharp interface method for SPH

Based on our previous sharp interface method (SIM) for smoothed particle hydrodynamics (SPH) [Zhang M and Deng X-L. A sharp interface method for SPH. Journal of Computational Physics 2015; 302 469–484], a robust SIM for SPH is developed. According to Lagrangian nature of SPH, the interface is located between two types of SPH particles. It is different from the previous method, in which the initial interface is defined by the user. The main consequence of the interface definition in the current method is that minimum value of level set function is around half the initial inter-particle distance. Therefore the calculation of level set function becomes more robust. The interface status is determined by jump conditions at the interface. Then the interface status is extended to the ghost fluid particles. Various benchmark tests are given to show the performance of the robust SIM for SPH. Comparing with the previous SIM for SPH, current method is more robust and accurate in the simulation of low-speed multiphase flows of high density ratios with clear interface.

Adaptive Cartesian cut-cell sharp interface method (aC3SIM) for three-dimensional multi-phase flows

A sharp interface method has been developed for treating interfacial discontinuities in compressible multi-phase fluids on three-dimensional Cartesian cut-cell grids. The evolution of the interfacial discontinuities in the Cartesian grid is captured by the level-set method. The intersections between interfacial fronts and Cartesian grids are interpolated using level-set function values at the vertices of Cartesian grids. Triangular surfaces are then constructed on the interfacial fronts. A novel cell merge method is used for complex topological changes. Jump conditions across discontinuous interfaces are enforced by reconstruction of interfacial flow variables using a constrained least-squares method. The inviscid flux across internal faces of the same fluid is calculated by the local Lax–Friedrichs flux. Manufactured solutions for interfaces are suggested for validation of the reconstruction method. Laplace’s law test results show that the present method drastically reduces the parasite currents compared to conventional interface treatment methods. Bubble rise problems also show the validity and accuracy of the proposed sharp interface method for immiscible two-phase fluids.

Development of new finite volume schemes on unstructured triangular grid for simulating the gas–liquid two‐phase flow

SummaryThis paper presents a coupled finite volume inner doubly iterative efficient algorithm for linked equations (IDEAL) with level set method to simulate the incompressible gas–liquid two‐phase flows with moving interfaces on unstructured triangular grid. The finite volume IDEAL method on a collocated grid is employed to solve the incompressible two‐phase Navier–Stokes equations, and the level set method is used to capture the moving interfaces. For the sake of mass conservation, an effective second‐order accurate finite volume scheme is developed to solve the level set equation on triangular grid, which can be implemented much easier than the classical high‐order level set solvers. In this scheme, the value of level set function on the boundary of control volume is approximated using a linear combination of a high‐order Larangian interpolation and a second‐order upwind interpolation. By the rotating slotted disk and stretching and shrinking of a circular fluid element benchmark cases, the mass conservation and accuracy of the new scheme is verified. Then the coupled method is applied to two‐phase flows, including a 2D bubble rising problem and a 2D dam breaking problem. The computational results agree well with those reported in literatures and experimental data. Copyright © 2015 John Wiley & Sons, Ltd.

New optimization method for steered fiber composites using the level set method

Advanced fiber placement (AFP) composite manufacturing technology offers a means to tailor composite fibers for complex loading environments and significantly improve the overall structural efficiency. This paper introduces a new method to optimize the continuously varying fiber paths for AFP using a level set method. The paths of the fibers are defined by constant level set function values, describing a series of continuous equally spaced fiber paths. The sensitivity of the structural compliance to a change in level set function definition of the fiber path is derived. The sensitivities are used to optimize the level set defined fiber paths to minimize structural compliance, while maintaining the continuous fiber paths and producing a solution that can be manufactured using AFP. The optimization method is demonstrated in three numerical studies.

Novelty detection using level set methods.

This paper presents a level set boundary description (LSBD) approach for novelty detection that treats the nonlinear boundary directly in the input space. The proposed approach consists of level set function (LSF) construction, boundary evolution, and termination of the training process. It employs kernel density estimation to construct the LSF of the initial boundary for the training data set. Then, a sign of the LSF-based algorithm is proposed to evolve the boundary and make it fit more tightly in the data distribution. The training process terminates when an expected fraction of rejected normal data is reached. The evolution process utilizes the signs of the LSF values at all training data points to decide whether to expand or shrink the boundary. Extensive experiments are conducted on benchmark data sets to evaluate the proposed LSBD method and compare it against four representative novelty detection methods. The experimental results demonstrate that the novelty detector modeled with the proposed LSBD can effectively detect anomalies.

History matching of facies distribution with varying mean lengths or different principle correlation orientations

Abstract The reservoir rock usually consists of several distinct facies. And, due to the different formation environments, the mean length of each facies could be regionally varying inside a layer or between different layers and the principle correlation orientations could be different between different facies. In this study, we investigate the history matching problem of facies distribution with varying mean lengths or different principle correlation orientations. The reparameterization scheme proposed by Chang et al. (2010) is adopted here to parameterize the facies distribution. Here we mainly investigate the establishment of the representing node system for the complex facies distribution considered and test its sensitivity. For the cases with isotropic correlation but regionally varying mean lengths, we propose a spatially non-uniform representing node system to suitably capture the regional information. For the cases with different principle correlation orientations, we propose a multiple representing node systems to suitably preserve the characteristic of each facies. Through the reparameterization, the level set function values generated for the representing nodes are the model parameters of the Ensemble Kalman filter (EnKF) and then be updated in the data assimilation process. Three synthetic cases are set up to test the performance of the proposed method, and the numerical results show that the main features of the reference fields can be captured by the realizations or the probabilistic map. The proposed method is shown to be flexible for the history matching problem of facies distribution with varying mean lengths or different principle correlation orientations.

A Kriging-Interpolated Level-Set Approach for Structural Topology Optimization

Level-set approaches are a family of domain classification techniques that rely on defining a scalar level-set function (LSF), then carrying out the classification based on the value of the function relative to one or more thresholds. Most continuum topology optimization formulations are at heart, a classification problem of the design domain into structural materials and void. As such, level-set approaches are gaining acceptance and popularity in structural topology optimization. In conventional level set approaches, finding an optimum LSF involves solution of a Hamilton-Jacobi system of partial differential equations with a large number of degrees of freedom, which in turn, cannot be accomplished without gradients information of the objective being optimized. A new approach is proposed in this paper where design variables are defined as the values of the LSF at knot points, then a Kriging model is used sto interpolate the LSF values within the rest of the domain so that classification into material or void can be performed. Perceived advantages of the Kriging-interpolated level-set (KLS) approach include alleviating the need for gradients of objectives and constraints, while maintaining a reasonable number of design variables that is independent from the mesh size. A hybrid genetic algorithm (GA) is then used for solving the optimization problem(s). An example problem of a short cantilever is studied under various settings of the KLS parameters in order to infer the best practice recommendations for tuning the approach. Capabilities of the approach are then further demonstrated by exploring its performance on several test problems.

History matching of fracture distributions by ensemble Kalman filter combined with vector based level set parameterization

Abstract For fractured reservoirs with unknown fracture distributions, the characteristics of fracture distributions are crucial for determining their production behaviors. Traditional history matching methods are not appropriate because the pixel-based rock property fields are usually highly non-Gaussian. In this work, a method that combines a vector-based level set reparameterization technique and the ensemble Kalman filter (EnKF) for estimating fracture distributions of two-dimensional reservoir model is presented. In the parameterization process, we first set up a group of representing nodes. The value of level set function on each node is assigned using Gaussian random number, and the sign of the function value indicates whether there is a fracture starting from the node or not. If there exists a linear fracture in a two dimensional space, the fracture would be characterized by the fracture length and orientation. Thus, the fracture distribution of the reservoir domain can be represented by a representing node system with a vector of three components on each node, which are the level set function, the fracture length, and the fracture angle. In the data assimilation process, these parameters are updated via EnKF scheme as the model parameters of the state vector. Two dimensional examples of water flooding in fractured reservoirs are set up to demonstrate the proposed method. It is shown that the method is effective to capture the main features of the fracture distributions in the reference fields. The matches of production data also improve significantly after updating.

A gradient-augmented level set method with an optimally local, coherent advection scheme

The level set approach represents surfaces implicitly, and advects them by evolving a level set function, which is numerically defined on an Eulerian grid. Here we present an approach that augments the level set function values by gradient information, and evolves both quantities in a fully coupled fashion. This maintains the coherence between function values and derivatives, while exploiting the extra information carried by the derivatives. The method is of comparable quality to WENO schemes, but with optimally local stencils (performing updates in time by using information from only a single adjacent grid cell). In addition, structures smaller than the grid size can be located and tracked, and the extra derivative information can be employed to obtain simple and accurate approximations to the curvature. We analyze the accuracy and the stability of the new scheme, and perform benchmark tests.

Interior sphere property for level sets of the value function of an exit time problem

We consider an optimal control problem for a system of the form ˙ x = f(x,u), with a running cost L. We prove an interior sphere property for the level sets of the corresponding value function V. From such a property we obtain a semiconcavity result for V , as well as perimeter estimates for the attainable sets of a symmetric control system. Mathematics Subject Classification. 93B03, 49L20, 49L25

Level Sets of the Value Function in Linear Differential Games with Elliptical Vectograms

Examples of linear differential games with fixed terminal time and elliptical constraints on the players controls are considered. These examples are interesting because some level sets of value function have narrow “throats”. Presence of such throats demands extremely precise computations during the numerical construction of the level sets.

Level Sets of Value Function and Singular Surfaces in Linear Differential Games

The paper deals with an algorithm of construction of level sets in linear differential games with fixed terminal time and convex payoff function depending on two components of the phase vector. Into this algorithm, the block for detection of singular points on the border of level set is included. Singular points give singular lines on the border of level set. Singular surfaces in the game space are collected using singular lines as skeleton. Examples of numerically calculated level sets and singular surfaces are represented.

Special Aspects of Convex Hull Constructing in Linear Differential Games of Small Dimension

The backward constructions for building the level sets of value function of differential games are discussed. Special aspects are outlined which can be used in fast algorithms. The examples of computer calculations are presented.