Parametric Level Set Method Research Articles

A Study on Basis Functions of the Parameterized Level Set Method for Topology Optimization of Continuums

Abstract In recent years, the parameterized level set method (PLSM), which rests on radial basis functions in most early work, has gained growing attention in structural optimization. However, little work has been carried out to investigate the effect of the basis functions in the parameterized level set method. This paper examines the basis functions of the parameterized level set method, including radial basis functions, B-spline functions, and shape functions in the finite element method (FEM) for topology optimization of continuums. The effects of different basis functions in the PLSM are examined by analyzing and comparing the required storage, convergence speed, computational efficiency, and optimization results, with the benchmark minimum compliance problems subject to a volume constraint. The linear basis functions show relatively satisfactory overall performance. Besides, several schemes to boost computational efficiency are proposed. The study on examples with unstructured 2D and 3D meshes can also be considered as a tentative investigation of prospective possible commercial applications of this method.

Three-dimensional topology optimization of auxetic metamaterial using isogeometric analysis and model order reduction

We present an approach for designing material micro-structures by using isogeometric analysis and parameterized level set method. Design variables, which are level set values associated with control points, are updated from the optimizer and represent the geometry of the unit cell. The computational efficiency is further improved in each iteration by employing reduced order modeling when solving linear systems of the equilibrium equations. We construct a reduced basis by reusing computed solutions from previous optimization steps, and a much smaller linear system of equations is solved on the reduced basis. Two- and three-dimensional numerical results show the effectiveness of the topology optimization algorithm coupled with the reduced basis approach in designing metamaterials.

Design of Self-Expanding Auxetic Stents Using Topology Optimization.

Implanting stents is the most efficient and minimally invasive technique for treating coronary artery diseases, but the risks of stent thrombosis (ST) and in-stent restenosis (IRS) hamper the healing process. There have been a variety of stents in market but dominated by ad hoc design motifs. A systematic design method that can enhance deliverability, safety and efficacy is still in demand. Most existing designs are focused on patient and biological factors, while the mechanical failures related to stenting architectures, e.g., inadequate stent expansion, stent fracture, stent malapposition and foreshortening, are often underestimated. With regard to these issues, the self-expanding (SE) stents may perform better than balloon-expandable (BE) stents, but the SE stents are not popular in clinic practice due to poor deliverability, placement accuracy, and precise match of the stent size and shape to the vessel. This paper addresses the importance between stent structures and clinic outcomes in the treatment of coronary artery disease. First, a concurrent topological optimization method will be developed to systematically find the best material distribution within the design domain. An extended parametric level set method with shell elements is proposed in the topology optimization to ensure the accuracy and efficiency of computations. Second, the auxetic metamaterial with negative Poisson’s ratio is introduced into the self-expanding stents. Auxetics can enhance mechanical properties of structures, e.g., fracture toughness, indentation and shear resistance and vibration energy absorption, which will help resolve the drawbacks due to the mechanical failures. Final, the optimized SE stent is numerically validated with the commercial software ANSYS and then prototyped using additive manufacturing techniques. Topological optimization gives a rare opportunity to exploiting the unique advantages of additive manufacturing. Hence, the topologically optimized auxetic architectures will provide a new solution for developing novel stenting structures, especially conductive to self-expanding SE stents. The new design will overcome the limitations of conventional SE stents associated with mechanical structures while maintain their valuable features, to help reduce the occurrence of ST and ISR and benefit the clinic practice in treating coronary heart disease.

Shape detection of multiple subsurface cavities by particle filtering with elastic wave propagation

SummaryA numerical technique for detecting the number and shape of subsurface cavities is presented, applying the particle filter and the parametric level set method to elastic wave propagation under the ground. When subsurface cavities exist, the elastic wave propagating in the ground is reflected at the boundary faces of the cavities. Observing the velocity of the reflection wave at the surface of a ground that includes multiple cavities and parameterizing the shape of the cavities by the parametric level function, both the number and the shape of the cavities can be identified by the particle filter. Numerical experimentation for detecting multiple cavities is conducted with synthetic observation data. The results show that the proposed technique enables the number of cavities to be identified by the number of peaks in the posterior probabilistic distribution function and solves geometric inverse problems by estimating the shape of the cavities through the parameter identification of the level set function.

Topology optimization under design-dependent loads with the parameterized level-set method based on radial-basis functions

Based on the parameterized level-set method using radial basis functions, a topology optimization method is proposed that can account for design-dependent loads. First, a mathematical model of the optimization problem is established with the structural compliance minimization as the objective function and the structural volume fraction as the design constraint. By converting the line integrals into volume integrals, the design-dependent loads imposed on the structure outline described by the zero-level set are converted, and a relatively easy boundary detection method is used. Subsequently, an updating strategy of the Lagrange multiplier is proposed to make the transition of the objective function smooth during the convergence process. Finally, the effectiveness and efficiency of the proposed optimization method in solving design-dependent load problems can be shown through several classic numerical examples.

A design framework for gradually stiffer mechanical metamaterial induced by negative Poisson's ratio property

Gradually stiffer (GS) mechanical metamaterial is a novel kind of adaptive structures under nonlinear varying loads for cushioning and vibration damping in engineering. These man-made metamaterials in applications are usually tailored by trial-and-error. This study develops a design framework that integrates the topology optimization, parametric design and compression experiment for the GS mechanical metamaterials induced by the negative Poisson's ratio (NPR) property. Firstly, a parametric level set method is incorporated with the numerical homogenization method to topologically optimize a series of microstructural unit cells with different NPRs. Then, the nonlinear static analysis is respectively applied to different unit cells to check their GS behavior. Secondly, typical microstructures with GS property are selected and simplified to perform the subsequent parametric design to analyze the expected property. Thirdly, the obtained mechanical metamaterials are fabricated by using the additive manufacturing (AM). GS properties of the obtained designs are analyzed and verified by both the simulation and compression experiments. Results show that we could obtain the GS mechanical metamaterial based on NPR properties for their similar characteristics. This study demonstrates that the proposed design framework is effective to tailor innovative layouts of the GS mechanical metamaterials.

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 parameterized level set method combined with polygonal finite elements in topology optimization

Level set-based topology optimization methods have received increased attention in the last decade due to the ease with which they can handle topology changes by splitting and merging boundaries. However, most implementations are limited to box-like design domains and cannot easily handle irregular domains that are common in practical engineering applications. In this paper, a modified level set model, which is parameterized and updated with radial basis functions (RBFs), is introduced to solve topology optimization problems in complex design domains. Because RBFs are not necessarily located in a structured manner, the parametrized level set method can be combined effortlessly with unstructured polygonal finite element meshes to easily handle complex design domains and also achieve the high accuracy characteristic of polygonal discretizations. Furthermore, the nucleation capability of the proposed approach along with an approximate reinitialization is shown to obtain more flexible topology changes of the optimal design, and smooth boundaries can be obtained without introducing smoothing scheme. Several numerical examples illustrate the effectiveness of the proposed model.

Maximizing natural frequencies of inhomogeneous cellular structures by Kriging-assisted multiscale topology optimization

This paper proposes a Kriging-assisted multiscale topology optimization method for maximizing natural frequencies of inhomogeneous cellular structures, where spatially-varying microstructural configurations and their macroscopic distribution are simultaneously optimized. At the beginning, under macroscopic boundary conditions for a cellular structure, the configurations of multiple prototype microstructures are topologically optimized by the parametric level set method (PLSM) combined with the numerical homogenization approach. A kinematical connective constraint is considered to ensure the connectivity between adjacent prototype microstructures. Then, a shape interpolation method is adopted to interpolate shapes of the prototype microstructures, so as to generate a series of sample microstructures. Based on these samples, Kriging metamodels are constructed to predict the effective property of each microstructure within the macrostructure. Finally, the variable thickness sheet (VTS) method is applied to optimize the material distribution pattern at macroscale for maximizing the natural frequency of the cellular structure, where an efficient mode-tracking strategy based on modal assurance criterion (MAC) is employed to track the target mode accurately. Numerical examples are provided to test the performance of the proposed method in natural frequency optimization of cellular structures. The results indicate that the multiscale cellular structures obtained by the proposed method show higher natural frequency compared with the monoscale macrostructural and microstructural designs.

A multi-discretization scheme for topology optimization based on the parameterized level set method

In the framework of the parameterized level set method, the structural analysis and topology representation can be implemented in a decoupling way. A parameterized level set function, typically, using radial basis functions (RBFs), is a linear combination of a set of prescribed RBFs and coefficients. Once the coefficients are determined, the theoretical level set function is determined. Exploiting this inherent property, we propose a multi-discretization method based on the parameterized level set method. In this approach, a coarse discretization is applied to do the structural analysis whereas another dense discretization is employed to represent the structure topology. As a result, both efficient analysis and high-resolution topological design are available. Note that the dense discretization only accounts for a more precise and smooth description of the theoretical level set function rather than introduce extra design freedom or incur interference to structural analysis or the optimization process. In other words, this decoupling way will not add to the computational burden of structural analysis or result in non-uniqueness of converged results for a particular analysis setting. Numerical examples in both two-dimension and three-dimension show effectiveness and applicability of the proposed method.

Multimodal 3D Shape Reconstruction under Calibration Uncertainty Using Parametric Level Set Methods

We consider the problem of 3D shape reconstruction from multimodal data, given uncertain calibration parameters. Typically, 3D data modalities can come in diverse forms such as sparse point sets, volumetric slices, and 2D photos. To jointly process these data modalities, we exploit a parametric level set method that utilizes ellipsoidal radial basis functions. This method not only allows us to analytically and compactly represent the object; it also confers on us the ability to overcome calibration-related noise that originates from inaccurate acquisition parameters. This essentially implicit regularization leads to a highly robust and scalable reconstruction, surpassing other traditional methods. In our results we first demonstrate the ability of the method to compactly represent complex objects. We then show that our reconstruction method is robust both to a small number of measurements and to noise in the acquisition parameters. Finally, we demonstrate our reconstruction abilities from diverse modalities such as volume slices obtained from liquid displacement (similar to CT scans and X-rays) and visual measurements obtained from shape silhouettes as well as point clouds.

B-Spline Level Set Method for Shape Reconstruction in Electrical Impedance Tomography.

A B-spline level set (BLS) based method is proposed for shape reconstruction in electrical impedance tomography (EIT). We assume that the conductivity distribution to be reconstructed is piecewise constant, transforming the image reconstruction problem into a shape reconstruction problem. The shape/interface of inclusions is implicitly represented by a level set function (LSF), which is modeled as a continuous parametric function expressed using B-spline functions. Starting from modeling the conductivity distribution with the B-spline based LSF, we show that the shape modeling allows us to compute the solution by restricting the minimization problem to the space spanned by the B-splines. As a consequence, the solution to the minimization problem is obtained in terms of the B-spline coefficients. We illustrate the behavior of this method using simulated as well as water tank data. In addition, robustness studies considering varying initial guesses, differing numbers of control points, and modeling errors caused by inhomogeneity are performed. Both simulation and experimental results show that the BLS-based approach offers clear improvements in preserving the sharp features of the inclusions in comparison to the recently published parametric level set method.

Multiscale topology optimization for coated structures with multifarious-microstructural infill

In engineering, cellular structures are often manufactured with coating for protection or to improve certain functionalities. This paper concentrates on the design of coated structures consisting of an exterior solid shell and an inner base part filled by multifarious microstructures, and a novel multiscale topology optimization method is proposed. Firstly, a representation method and a material interpolation model are developed to describe a coated structure with multifarious-microstructural infill and define its properties, respectively. At macroscale, coating-base distribution is determined by the parametric level set method (PLSM) with re-initialization. To optimize structural performance at a computationally affordable cost, controllable kinds of microstructures are considered, and their spatial distribution over the whole base region is optimized by the ordered SIMP method with a threshold scheme. At microscale, the configurations of microstructures are generated by PLSM with the numerical homogenization method, in which the volume fraction limit values correspond to the design variables of the ordered SIMP method in macroscale. The compliance minimization problem subject to a material mass constraint is investigated, and sensitivity analysis is derived. Numerical examples are provided to demonstrate the effectiveness of the proposed method.

Topology Optimization of Micro-Structured Materials Featured with the Specific Mechanical Properties

Micro-structured materials consisting of an array of microstructures are engineered to provide the specific material properties. This present work investigates the design of cellular materials under the framework of level set, so as to optimize the topologies and shapes of these porous material microstructures. Firstly, the energy-based homogenization method (EBHM) is applied to evaluate the material effective properties based on the topology of the material cell, where the effective elasticity property is evaluated by the average stress and strain theorems. Secondly, a parametric level set method (PLSM) is employed to optimize the microstructural topology until the specific mechanical properties can be achieved, including the maximum bulk modulus, the maximum shear modulus and their combinations, as well as the negative Poisson’s ratio (NPR). The complicated topological shape optimization of the material microstructure has been equivalent to evolve the sizes of the expansion coefficients in the interpolation of the level set function. Finally, several numerical examples are fully discussed to demonstrate the effectiveness of the developed method. A series of new and interesting material cells with the specific mechanical properties can be found.

The parameterized level set method for structural topology optimization with shape sensitivity constraint factor

In recent years, the parameterized level set method (PLSM) has attracted widespread attention for its good stability, high efficiency and the smooth result of topology optimization compared with the conventional level set method. In the PLSM, the radial basis functions (RBFs) are often used to perform interpolation fitting for the conventional level set equation, thereby transforming the iteratively updating partial differential equation (PDE) into ordinary differential equations (ODEs). Hence, the RBFs play a key role in improving efficiency, accuracy and stability of the numerical computation in the PLSM for structural topology optimization, which can describe the structural topology and its change in the optimization process. In particular, the compactly supported radial basis function (CS-RBF) has been widely used in the PLSM for structural topology optimization because it enjoys considerable advantages. In this work, based on the CS-RBF, we propose a PLSM for structural topology optimization by adding the shape sensitivity constraint factor to control the step length in the iterations while updating the design variables with the method of moving asymptote (MMA). With the shape sensitivity constraint factor, the updating step length is changeable and controllable in the iterative process of MMA algorithm so as to increase the optimization speed. Therefore, the efficiency and stability of structural topology optimization can be improved by this method. The feasibility and effectiveness of this method are demonstrated by several typical numerical examples involving topology optimization of single-material and multi-material structures.

Topology optimization of 2-D mechanical metamaterials using a parametric level set method combined with a meshfree algorithm

Two-dimensional (2-D) micro-architectured mechanical metamaterials are designed using a topology optimization approach that integrates a parametric level set method (PLSM) with a meshfree method based on compactly supported radial basis functions (CS-RBF). The PLSM is employed as the optimization algorithm to achieve desired microstructures with targeted material properties. The effective elastic properties, including the bulk modulus, shear modulus and Poisson’s ratio, are predicted using a strain energy-based homogenization method and the CS-RBF meshfree algorithm. Two sets of optimizations are implemented: one is for the case of a single solid material, and the other is for the case of two solid materials, each involving an additional void phase. Three numerical examples are provided for each case with the same optimization objectives: maximizing the effective bulk modulus, maximizing the effective shear modulus, and minimizing the effective Poisson’s ratio under given volume fraction constraints. The numerical results reveal that the newly proposed approach can generate smooth topological boundaries and optimal microstructures. In particular, the current method can topologically optimize auxetic metamaterials with a negative Poisson’s ratio. It can also be extended to design other periodic metamaterials, including those with a negative coefficient of thermal expansion or frequency bandgaps.

A new multiscale topology optimization method for multiphase composite structures of frequency response with level sets

This paper proposes a new multiscale topology optimization method for the concurrent design of multiphase composite structures under a certain range of excitation frequencies. Distinguished from the existed studies, a general concurrent design formulation for the dynamic composite structures with more than two material phases is developed. The macrostructureand its microstructures with multiple material phases are optimized simultaneously. The integral of the dynamic compliances over an interval of frequencies is formulated as the optimization objective, so as to minimize the frequency response within the concerned excitation range. The effective properties of the multiphase microstructures are evaluated by using the numerical homogenization method, which actually serves as a link to bridge the macro and micro finite element analyses. Furthermore, to describe the boundaries of multiple material phases for the microstructure, a parametric color level set method (PCLSM) is developed by using an efficient interpolation scheme. In this way, L level set functions can represent at most 2L material phases without any overlaps. Moreover, these “color” level sets are updated by directly using the well-established gradient-based algorithm, which can greatly facilitate the proposed method to solve the multi-material optimizations with multiple design constraints. Several 2D and 3D numerical examples are used to demonstrate the effectiveness of the proposed method in the concurrent design of the dynamic composite structures under the excitation frequency ranges.

Fully parallel level set method for large-scale structural topology optimization

To realize large-scale or high-resolution structural topology optimization design, a fully parallel parameterized level set method with compactly supported radial basis functions (CSRBFs) is developed based on both the uniform and non-uniform structured meshes. In this work, the whole computation process is parallelized, including mesh generation, sensitivity analysis, calculation and assembly of the element stiffness matrices, solving of the structural state equation, parameterization and updating of the level set function, and output of the computational results during the optimization iterations. In addition, some typical numerical examples, in which the calculation scale is up to 7 million 8-node hexahedral elements, are carried out for verifying the effectiveness of the proposed method. Finally, the computing time is also analyzed in detail. It is found that: (1) In the optimized structures, the thin sheet-like components gradually replace the truss-like ones when refining the mesh, (2) the parameterization process of the level set function will become fast as long as the non-uniformity of mesh is not very high and the supported radius of CSRBF is small enough, and (3) more than 80% of the total computing time is always consumed for solving the structural state equation during the finite element analysis (FEA).

Parametric Topology Optimization Toward Rational Design and Efficient Prefabrication for Additive Manufacturing

The significant advance in the boosted fabrication speed and printing resolution of additive manufacturing (AM) technology has considerably increased the capability of achieving product designs with high geometric complexity and provided tremendous potential for mass customization. However, it is also because of geometric complexity and large quantity of mass-customized products that the prefabrication (layer slicing, path planning, and support generation) is becoming the bottleneck of the AM process due to the ever-increasing computational cost. In this paper, the authors devise an integrated computational framework by synthesizing the parametric level set-based topology optimization method with the stereolithography (SLA)-based AM process for intelligent design and manufacturing of multiscale structures. The topology of the design is optimized with a distance-regularized parametric level set method considering the prefabrication computation. With the proposed framework, the structural topology optimization not only can create single material structure designs but also can generate multiscale, multimaterial structures, offering the flexibility and robustness of the structural design that the conventional methods could not provide. The output of the framework is a set of mask images that can be directly used in the AM process. The proposed approach seamlessly integrates the rational design and manufacturing to reduce the numerical complexity of the computationally expensive prefabrication process. More specifically, the prefabrication-friendly design and optimization procedure are devised to drastically eliminate the redundant computations in the traditional framework. Two test examples, including a free-form 3D cantilever beam and a multiscale meta-structure, are utilized to demonstrate the performance of the proposed approach. Both the simulation and experimental results verified that the new rational design could significantly reduce the prefabrication computation cost without affecting the original design intent or sacrificing the original functionality.

Concurrent optimization of structural topology and infill properties with a CBF-based level set method

In this paper, a parametric level-set-based topology optimization framework is proposed to concurrently optimize the structural topology at the macroscale and the effective infill properties at the micro/meso scale. The concurrent optimization is achieved by a computational framework combining a new parametric level set approach with mathematical programming. Within the proposed framework, both the structural boundary evolution and the effective infill property optimization can be driven by mathematical programming, which is more advantageous compared with the conventional partial differential equation-driven level set approach. Moreover, the proposed approach will be more efficient in handling nonlinear problems with multiple constraints. Instead of using radial basis functions (RBF), in this paper, we propose to construct a new type of cardinal basis functions (CBF) for the level set function parameterization. The proposed CBF parameterization ensures an explicit impose of the lower and upper bounds of the design variables. This overcomes the intrinsic disadvantage of the conventional RBF-based parametric level set method, where the lower and upper bounds of the design variables oftentimes have to be set by trial and error. A variational distance regularization method is utilized in this research to regularize the level set function to be a desired distanceregularized shape. With the distance information embedded in the level set model, the wrapping boundary layer and the interior infill region can be naturally defined. The isotropic infill achieved via the mesoscale topology optimization is conformally fit into the wrapping boundary layer using the shape-preserving conformal mapping method, which leads to a hierarchical physical structure with optimized overall topology and effective infill properties. The proposed method is expected to provide a timely solution to the increasing demand for multiscale and multifunctional structure design.