Parametric Level Set Method Research Articles

Dynamic multiscale topology optimization for multi-regional micro-structured cellular composites

In this paper, a new dynamic multiscale topology optimization method for cellular composites with multi-regional material microstructures is proposed to improve the structural performance. Firstly, a free-material distribution optimization method (FMDO) is developed to generate the overall configuration for the discrete element densities distributed within a multi-regional pattern. The macrostructure is divided into several sub regions, and each of them consists of a number of elements but with the same densities. Secondly, a dynamic topology optimization formulation is developed to perform the concurrent design of the macrostructure and material microstructures, subject to the multi-regional distributed element densities. A parametric level set method is employed to optimize the topologies of the macrostructure and material microstructures, with the effective macroscopic properties evaluated by the homogenization. In the numerical implementation, the quasi-static Ritz vector (QSRV) method is incorporated into the finite element analysis so as to reduce the computational cost in numerical analysis, and some kinematical connectors are introduced to make sure the connectivity between adjacent material microstructures. Finally, 2D and 3D numerical examples are tested to demonstrate the effectiveness of the proposed dynamic multiscale topology optimization method for the material-structural composites.

Concurrent topology optimization for cellular structures with nonuniform microstructures based on the kriging metamodel

This paper proposes a novel multiscale concurrent topology optimization for cellular structures with continuously varying microstructures in space to obtain a superior structural performance at an affordable computation cost. At microscale, multiple prototype microstructures are topologically optimized to represent all the microstructures within macrostructure by incorporating a numerical homogenization approach into a parametric level set method (PLSM), whose connectivity is guaranteed by a kinematical connective constraint approach. A shape interpolation technology is developed to map these optimized prototype microstructures and generate a series of nonuniform microstructures, which are considered as sample points and used to construct a kriging metamodel. The built kriging metamodel is then employed to predict the effective properties of all the nonuniform microstructures within macrostructure. At macroscale, the variable thickness sheet (VTS) method is employed to generate an overall free material distribution patterns using the predicted effective properties of all the nonuniform microstructures. With the help of shape interpolation technology, all the nonuniform microstructures within macrostructure are well connected with each other due to the similar topological features at their interfaces. Using the proposed method, the macrostructural topology as well as the locations and configurations of the spatially varying nonuniform microstructures can be simultaneously optimized to ensure a sufficiently large multiscale design space. Numerical examples are provided to demonstrate the validity and advantages of the proposed method.

An improved parametric level set method for structural frequency response optimization problems

In conventional parametric level set methods, the compactly supported radial basis functions (CSRBF) are used to approximate the level set function due to their unique properties, such as the sparsity of the interpolation matrix. The CSRBFs only consider the contributions of knots within a narrow sub-region, which sacrifices accuracy for efficiency in the interpolation. However, the accuracy loss in the CSRBF-based method may prolong the iteration and gradually lead the topology optimization towards a worse local optimum or even an unfeasible design, especially when the allowable material usage in the design domain is relatively low. This will significantly affect the performance of the optimization method. This paper proposes an improved parametric level set method (iPLSM), which is more efficient and effective in topology optimization designs. In this method, the Gaussian radial basis function with global support is used to parameterize the level set surface, to ensure a high numerical accuracy due to the consideration of all interpolation knots in the global domain. Then, a discrete wavelet transform scheme is incorporated into the parametric form to compress the full interpolation matrix and save the computational cost. The proposed method is applied to both the global and local frequency response optimization problems under wide excitation frequency ranges, to validate its efficiency and effectiveness.

A level set–based method for stress‐constrained multimaterial topology optimization of minimizing a global measure of stress

SummaryIn multimaterial topology optimization of minimizing a global measure of stress, the maximum stresses in different materials may not satisfy the strength design requirements simultaneously if stress constraints for different materials are not considered. In this paper, a level set–based method is presented to handle the stress‐constrained multimaterial topology optimization of minimizing a global stress measure. Specifically, a multimaterial level set model is adopted to describe the structural topology, and a stress interpolation scheme is introduced for stress evaluation. Then, a stress penalty‐based topology optimization model is presented. Meanwhile, an adaptive adjusting scheme of the stress penalty factor is employed to improve the control of the local stress level. To solve the stress‐constrained multimaterial topology optimization problem minimizing the global measure of stress, the parametric level set method is employed, and the sensitivity analysis is carried out. Numerical examples are provided to demonstrate the effectiveness of the presented method. Results indicate that multimaterial structures with optimized global stress can be gained, and stress constraints for different materials can be satisfied simultaneously.

An Efficient Topology Optimization Method for Structures with Uniform Stress

This paper focuses on two kinds of bi-objective topology optimization problems with uniform-stress constraints: compliance-volume minimization and local frequency response–volume minimization problems. An adaptive volume constraint (AVC) algorithm based on an improved bisection method is proposed. Using this algorithm, the bi-objective uniform-stress-constrained topology optimization problem is transformed into a single-objective topology optimization problem and a volume-decision problem. The parametric level set method based on the compactly supported radial basis functions is employed to solve the single-objective problem, in which a self-organized acceleration scheme based on shape derivative and topological sensitivity is proposed to adaptively adjust the derivative of the objective function and the step length during the optimization. To solve the volume-decision problem, an improved bisection method is proposed. Numerical examples are tested to illustrate the feasibility and effectiveness of the self-organized acceleration scheme and the AVC algorithm based on the improved bisection method. An extended application to the bi-objective stress-constrained topology optimization of a structure with stress concentration is also presented.

Topology optimization for multiscale design of porous composites with multi-domain microstructures

This paper proposes a new multiscale topology optimization method for the design of porous composites composed of the multi-domain material microstructures considering three design elements: the topology of the macrostructure, the topologies of multiple material microstructures and their overall distribution in the macrostructure. The multiscale design involves two optimization stages: the free material distribution optimization and the concurrent topology optimization. Firstly, the variable thickness sheet (VTS) method with the regularization mechanism is used to generate multiple element density distributions in the macro design domain. Hence, different groups of elements with the identical densities can be uniformly arranged in their corresponding domains, and each domain in the space will be periodically configured by a unique representative microstructure. Secondly, with the discrete material distributions achieved in the macro domain, the topology of the macrostructure and topologies of multiple representative microstructures are concurrently optimized by a parametric level set method combined with the numerical homogenization method. Finally. Several 2D and 3D numerical examples are provided to demonstrate the effectiveness of the proposed multiscale topology optimization method.

Multiscale concurrent topology optimization for cellular structures with multiple microstructures based on ordered SIMP interpolation

This paper presents a novel multiscale concurrent topology optimization method for cellular structures with various types of microstructures to obtain a superior structural performance at an affordable computation cost. At macroscale, different microstructures are regarded as different materials in a multi-material topology optimization. An ordered solid isotropic material with penalization (SIMP) interpolation model, which is efficient for solving multi-material topology optimization problems without introducing any new variables, is employed to generate an overall structure layout with predetermined multiple materials. At microscale, each macroscale element is viewed as an individual microstructure. All macroscale elements with the same material are represented by a unique microstructure. A parametric level set method (PLSM) is applied to evolve the shape and topology of the representative microstructures, whose effective properties are evaluated by a numerical homogenization method. The connectivity of adjacent microstructures is guaranteed by a kinematical connective constraint. Using the proposed method, the macrostructural topology as well as the locations and configurations of microstructures can be simultaneously optimized. Numerical examples are provided to test the effectiveness of the proposed method.

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.

A Parametric Level Set-Based Approach to Difference Imaging in Electrical Impedance Tomography.

This paper presents a novel difference imaging approach based on the recently developed parametric level set (PLS) method for estimating the change in a target conductivity from electrical impedance tomography measurements. As in conventional difference imaging, the reconstruction of conductivity change is based on data sets measured from the surface of a body before and after the change. The key feature of the proposed approach is that the conductivity change to be reconstructed is assumed to be piecewise constant, while the geometry of the anomaly is represented by a shape-based PLS function employing Gaussian radial basis functions (GRBFs). The representation of the PLS function by using GRBF provides flexibility in describing a large class of shapes with fewer unknowns. This feature is advantageous, as it may significantly reduce the overall number of unknowns, improve the condition number of the inverse problem, and enhance the computational efficiency of the technique. To evaluate the proposed PLS-based difference imaging approach, results obtained via simulation, phantom study, and in vivo pig data are studied. We find that the proposed approach tolerates more modeling errors and leads to a significant improvement in image quality compared with the conventional linear approach.

Parameterized level-set based topology optimization method considering symmetry and pattern repetition constraints

Symmetry and repetitive patterns are very common in both natural and artificial structures. Design with symmetry and pattern repetition (SPR) can be very useful in practical engineering. In this paper, we discuss a new optimal structural design method considering SPR constraints. The basic idea consists in combining a parameterized level-set method together with radial basis function (RBF) interpolation strategy, which allows for decoupling RBF knots from the finite element (FE) analysis mesh. The merits of the decoupled manner are that RBF knots can be mapped independently on regular or irregular design domain, and SPR constraints can be imposed by simply building a bijective relationship between RBF knots regardless of analysis mesh. Furthermore, due to parameterization of the level-set function, there are fewer numerical manipulations that need to be implemented than the conventional level-set method during the optimization process. Several numerical examples are presented to show feasibility and effectiveness of the proposed method.

An 88-line MATLAB code for the parameterized level set method based topology optimization using radial basis functions

This paper presents a compact and efficient 88-line MATLAB code for the parameterized level set method based topology optimization using radial basis functions (RBFs), which is applied to minimize the compliance of a two-dimensional linear elastic structure. This parameterized level set method using radial basis functions can maintain a relatively smooth level set function with an approximate re-initialization scheme during the optimization process. It also has less dependency on initial designs due to its capability in nucleation of new holes inside the material domain. The MATLAB code and simple modifications are explained in detail with numerical examples. The 88-line code included in the appendix is intended for educational purposes.

A Form Finding Method for Arch Bridges Using Parametric Level Set Method

Topology optimization has developed rapidly in the past three decades; as a creative and efficient optimization technique, it has been applied in engineering fields of aerospace and mechanical. However, there are a few attempts in bridge form design. In this paper, the parametric level set method is utilized to solve the form finding of arch bridges. The optimization model for minimizing the structural compliance under the volume constraint is built. Three numerical examples of form finding of arch bridges are studied. Results show that the optimal structures which have well‐distributed stress and smooth force transmission are almost identical with the actual forms of arch bridges. The optimal forms can be treated as alternatives in the preliminary design stage, and topology optimization has a bright prospect in form finding of arch bridges.

Topological shape optimization of 3D micro-structured materials using energy-based homogenization method

This paper proposes an effective method for the design of 3D micro-structured materials to attain extreme mechanical properties, which integrates the firstly developed 3D energy-based homogenization method (EBHM) with the parametric level set method (PLSM). In the 3D EBHM, a reasonable classification of nodes in periodic material microstructures is introduced to develop the 3D periodic boundary formulation consisting of 3D periodic boundary conditions, 3D boundary constraint equations and the reduced linearly elastic equilibrium equation. Then, the effective elasticity properties of material microstructures are evaluated by the average stress and strain theorems rather than the asymptotic theory. Meanwhile, the PLSM is applied to optimize microstructural shape and topology because of its positive characteristics, like the perfect demonstration of geometrical features and high optimization efficiency. Numerical examples are provided to demonstrate the advantages of the proposed design method. Results indicate that the optimized 3D material microstructures with expected effective properties are featured with smooth structural boundaries and clear interfaces.

Parametric shape and topology optimization: A new level set approach based on cardinal basis functions

SummaryThe parametric level set approach is an extension of the conventional level set methods for topology optimization. By parameterizing the level set function, level set methods can be directly coupled with mathematical programming to achieve better numerical robustness and computational efficiency. Moreover, the parametric level set scheme can not only inherit the primary advantages of the conventional level set methods, such as clear boundary representation and the flexibility in handling topological changes, but also alleviate some undesired features from the conventional level set methods, such as the need for reinitialization. However, in the existing radial basis function–based parametric level set method, it is difficult to identify the range of the design variables. Besides, the parametric level set evolution often struggles with large fluctuations during the optimization process. Those issues cause difficulties both in numerical stability and in material property mapping. In this paper, a cardinal basis function is constructed based on the radial basis function partition of unity collocation method to parameterize the level set function. The benefit of using cardinal basis function is that the range of the design variables can now be clearly specified as the value of the level set function. A distance regularization energy functional is also introduced, aiming to maintain the desired signed distance property during the level set evolution. With this desired feature, the level set evolution is stabilized against large fluctuations. In addition, the material properties mapped from the level set function to the finite element model can be more accurate.

Topology optimization for concurrent design of structures with multi-patch microstructures by level sets

This paper focuses on the novel concurrent design for cellular structures consisting of multiple patches of material microstructures using a level set-based topological shape optimization method. The macro structure is featured with the configuration of a cluster of non-uniformly distributed patches, while each patch hosts a number of identical material microstructures. At macro scale, a discrete element density based approach is presented to generate an overall structural layout involving different groups of discrete element densities. At micro scale, each macro element is regarded as an individual microstructure with a discrete intermediate density. Hence, all the macro elements with the same discrete densities (volume fractions) are represented by a unique microstructure. The representative microstructures corresponding to different density groups are topologically optimized by incorporating the numerical homogenization approach into a parametric level set method. The multiscale concurrent designs are integrated into a uniform optimization procedure, so as to optimize both topologies for the macrostructure and its microstructures, as well as locations of the microstructures in the design space. Numerical examples demonstrate that the proposed method can substantially improve the structural performance with an affordable computation and manufacturing cost.

Stress‐based multi‐material topology optimization of compliant mechanisms

AbstractIn this paper, a level‐set‐based method is presented to deal with the multi‐material topology optimization of compliant mechanisms with stress constraints. A novel stress‐based multi‐material topology optimization model of compliant mechanisms is proposed. In this model, the multi‐material level set topology description model and the separable stress interpolation scheme are adopted. The weighted sum method is used to deal with the multi‐objective optimization of the output displacement and compliance of compliant mechanisms. The penalty of stresses is also considered in the objective function to control the local stress level in different materials. To solve the optimization problem, the parametric level set method is employed, and the sensitivity analysis is conducted. Application of the method is demonstrated by 2 numerical examples. Results show that the multi‐material structures without undesirable de facto hinges can be obtained. The output displacement and compliance of the compliant mechanisms are optimized, and stress constraints in different materials are simultaneously satisfied.

A Parametric Level Set Method for Electrical Impedance Tomography.

This paper presents an image reconstruction method based on parametric level set (PLS) method using electrical impedance tomography. The conductivity to be reconstructed was assumed to be piecewise constant and the geometry of the anomaly was represented by a shape-based PLS function, which we represent using Gaussian radial basis functions (GRBF). The representation of the PLS function significantly reduces the number of unknowns, and circumvents many difficulties that are associated with traditional level set (TLS) methods, such as regularization, re-initialization and use of signed distance function. PLS reconstruction results shown in this article are some of the first ones using experimental EIT data. The performance of the PLS method was tested with water tank data for two-phase visualization and with simulations which demonstrate the most popular biomedical application of EIT: lung imaging. In addition, robustness studies of the PLS method w.r.t width of the Gaussian function and GRBF centers were performed on simulated lung imaging data. The experimental and simulation results show that PLS method has significant improvement in image quality compared with the TLS reconstruction.

A new method based on adaptive volume constraint and stress penalty for stress-constrained topology optimization

This paper focuses on the stress-constrained topology optimization of minimizing the structural volume and compliance. A new method based on adaptive volume constraint and stress penalty is proposed. According to this method, the stress-constrained volume and compliance minimization topology optimization problem is transformed into two simple and related problems: a stress-penalty-based compliance minimization problem and a volume-decision problem. In the former problem, stress penalty is conducted and used to control the local stress level of the structure. To solve this problem, the parametric level set method with the compactly supported radial basis functions is adopted. Meanwhile, an adaptive adjusting scheme of the stress penalty factor is used to improve the control of the local stress level. To solve the volume-decision problem, a combination scheme of the interval search and local search is proposed. Numerical examples are used to test the proposed method. Results show the lightweight design, which meets the stress constraint and whose compliance is simultaneously optimized, can be obtained by the proposed method.

Salt Reconstruction in Full-Waveform Inversion With a Parametric Level-Set Method

Seismic full-waveform inversion tries to estimate subsurface medium parameters from seismic data. Areas with subsurface salt bodies are of particular interest because they often have hydrocarbon reservoirs on their sides or underneath. Accurate reconstruction of their geometry is a challenge for current techniques. This paper presents a parametric level-set method for the reconstruction of salt-bodies in seismic full-waveform inversion. We split the subsurface model in two parts: a background velocity model and the salt body with known velocity but undetermined shape. The salt geometry is represented by a level-set function that evolves during the inversion. We choose radial basis functions to represent the level-set function, leading to an optimization problem with a modest number of parameters. A common problem with level-set methods is to fine tune the width of the level-set boundary for optimal sensitivity. We propose a robust algorithm that dynamically adapts the width of the level-set boundary to ensure faster convergence. Tests on a suite of idealized salt geometries show that the proposed method is stable against a modest amount of noise. We also extend the method to joint inversion of both the background velocity model and the salt-geometry.

Disjunctive Normal Parametric Level Set With Application to Image Segmentation.

Level set methods are widely used for image segmentation because of their convenient shape representation for numerical computations and capability to handle topological changes. However, in spite of the numerous works in the literature, the use of level set methods in image segmentation still has several drawbacks. These shortcomings include formation of irregularities of the signed distance function, sensitivity to initialization, lack of locality, and expensive computational cost, which increases dramatically as the number of objects to be simultaneously segmented grows. In this paper, we propose a novel parametric level set method called disjunctive normal level set (DNLS), and apply it to both two-phase (single object) and multiphase (multiobject) image segmentations. DNLS is a differentiable model formed by the union of polytopes, which themselves are created by intersections of half-spaces. We formulate the segmentation algorithm in a Bayesian framework and use a variational approach to minimize the energy with respect to the parameters of the model. The proposed DNLS can be considered as an open framework that allows the use of different appearance models and shape priors. Compared with the conventional level sets available in the literature, the proposed DNLS has the following major advantages: it requires significantly less computational time and memory, it naturally keeps the level set function regular during the evolution, it is more suitable for multiphase and local region-based image segmentations, and it is less sensitive to noise and initialization. The experimental results show the potential of the proposed method.