Parametric Level Set Method Research Articles

Defect identification in heat transfer problems using boundary data

The identification of internal defects has been widely focused and employed in the field of nondestructive evaluation (NDE), whereas its application in inverse heat transfer is often restricted by connectedness. Thus, in this paper, a novel identification method of multiple defects based on the parametric level set method (PLSM) and the Method of Moving Asymptotes (MMA) is proposed, where the connectedness limitation is released. In PLSM, the compactly supported radial basis functions (CSRBFs) are utilized to decouple the material related level set function, which endows the PLSM to be capable of forming high curvature portions and capturing local details of the defects. After that, the evolution of guessed level set can be tracked by updating the time-dependent expansion coefficients. Then, a least-square problem is defined to find the optimal coefficients that correspond to the actual defects. To obtain the controllable convergence and stabilize the solving of the ill-posed optimization problem, the MMA is employed to minimize the objective function. Furthermore, no requirement for the mesh reconstruction after the usage of ersatz material model, one also renders the sensitivity prone to compute. The efficiency and accuracy of the proposed method are further validated in several numerical examples by discussing the identification of single and multiple defects.

An interval-oriented dynamic robust topology optimization (DRTO) approach for continuum structures based on the parametric Level-Set method (PLSM) and the equivalent static loads method (ESLM)

An interval-oriented dynamic robust topology optimization (DRTO) approach for continuum structures based on the parametric Level-Set method (PLSM) and the equivalent static loads method (ESLM)

A robust topology optimization method considering bounded field parameters with uncertainties based on the variable time step parametric level-set method

In this paper, a robust topology optimization method considering bounded field parameters with uncertainties based on the variable time step parametric level-set method is proposed. Firstly, a variable time step strategy for level set function evolution based on the gradient of the level set function that uses radial basis function interpolation to realize the separation of time and space is developed, which can achieve better design results during topology optimization. Beyond that, the dimension reduction method and dimension-wise method based on the polynomials are used for characterization and quantification of bounded field parameters with uncertainties. Finally, the sensitivity of the robust optimization model is derived based on the shape derivative principle, which provides the basis for the implementation of gradient based optimization algorithm. Three examples illustrate the effectiveness, necessity and influence of important parameters of the robust topology optimization method considering bounded field parameters with uncertainties based on the variable time step parametric level-set method from different aspects.

Design of multiphase auxetic metamaterials by a parametric color level set method

This paper presents a systematic optimization design method for the multiphase auxetic metamaterials with different deformation mechanisms in both 2D and 3D scenarios. In this method, the parametric color level set (PCLS) is developed to describe different material phases in the microstructures, in which at most 2L materials phases can be precisely represented by only L description functions without any overlaps. Furthermore, clear and smooth material interfaces can be guaranteed in the design, and multiple material usage constraints can be efficiently handled by the well-established gradient-based algorithm. The shape derivative theory is introduced to analyze the design sensitivities for the optimization problem. The effective elasticity properties of the multiphase composites are evaluated by the numerical homogenization method under periodic boundary conditions. Various symmetric conditions are defined and enforced to induce the re-entrant and chiral patterns in both 2D and 3D metamaterials. Several numerical examples are provided to demonstrate the features of the proposed method in tailoring different types of multiphase auxetics. The multiphase metamaterials with two and more material phases are discussed. It is shown that the presented design method can be used to devise both the re-entrant and chiral auxetic metamaterials with excellent auxetic and stiffness properties.

Hybrid explicit–implicit topology optimization method for the integrated layout design of compliant mechanisms and actuators

Increasing the working stroke of compliant mechanisms in a limited space has always been an important topic in mechanics. One effective method is to develop a more fundamental design model that considers the multi-physical coupling characteristics of compliant mechanisms and actuators. This paper presents a new method for the integrated design of compliant mechanisms and piezoelectric actuators, which incorporates the projective transformation-based moving morphable components method with the parametric level set method (PMMC-PLS), based on the use of explicit and implicit topology optimization methods to drive the layout evolution of the embedded actuator and host structure, respectively. The extended finite element method (XFEM) is adopted to accurately capture the boundary of the multi-component system, thereby increasing the accuracy of the structural response and sensitivity analysis. To circumvent defacto hinges and thin wall features, a global manufacturability constraint is proposed by applying the minimum length scale control to the host structure and a non-overlap constraint to embedded actuators. Moreover, the output stiffness is considered to enhance the mechanical performance of the mechanism. The effectiveness of the proposed method is verified considering numerical examples.

A multiscale topological design method of geometrically asymmetric porous sandwich structures for minimizing dynamic compliance

Compared with conventional sandwich structures with two identical face sheets, geometrically asymmetric porous sandwich structures (GAPSSs) can achieve better dynamic performances under certain load and boundary condition, since they provide more choices for geometric design. However, current studies regarding the GAPSSs are mainly analytical- and experimental-based methods with predefined face sheet thicknesses and core configurations, few works investigate the GAPSSs by optimization method. This paper presents a multiscale topology optimization method of the GAPSSs, which is capable of designing both thicknesses of two solid face sheets at macroscale and configurations of porous core at microscale for minimizing dynamic compliance. Specifically, at macroscale, a variable thickness sheet method is employed to optimize the thicknesses of two solid face sheets. Then, at microscale, a parametric level-set method integrated with a numerical homogenization approach is applied to topologically optimize the configuration of the periodic unit cell. The method of moving asymptotes is adopted to update the design variables at both scales. Several 2D and 3D numerical examples are illustrated to show the effectiveness and advantages of the proposed method for designing the GAPSSs. The results indicate that the dynamic compliance of the optimized GAPSSs show superior advantages over the conventional sandwich structures with identical face sheet thicknesses and predefined lattice core.

A parameterized level set method for structural topology optimization based on reaction diffusion equation and fuzzy PID control algorithm

<abstract> <p>We propose a parameterized level set method (PLSM) for structural topology optimization based on reaction diffusion equation (RDE) and fuzzy PID control algorithm. By using the proposed method, the structural compliance minimization problem under volume constraints is studied. In this work, the RDE is used as the evolution equation of level set function, and the topological derivative of the material domain is used as the reaction term of the RDE to drive the evolution of level set function, which has little dependence on the initial design domain, and can generate holes in the material domain; the compactly supported radial basis function (CS-RBF) is used to interpolate the level set function and modify the RDE, which can improve the computational efficiency, and keep the boundary smooth in the optimization process. Meanwhile, the fuzzy PID control algorithm is used to deal with the volume constraints, so that the convergence process of the structure volume is relatively stable. Furthermore, the proposed method is applied to 3D structural topology optimization. Several typical numerical examples are provided to demonstrate the feasibility and effectiveness of this method.</p> </abstract>

CPU parallel-based adaptive parameterized level set method for large-scale structural topology optimization

CPU parallel-based adaptive parameterized level set method for large-scale structural topology optimization

A novel parametric level set method coupled with Tikhonov regularization for tomographic laser absorption reconstruction

• Clear topological boundary of the combustion reaction flow is obtained. • Variations of temperature and species concentration in the chamber are diagnosed. • The accuracy, robustness and noise immunity of proposed method are the best. • The diagnosis of reaction flow with a large boundary gradient is highly accurate. Temperature and concentration distributions are essential parameters for understanding physicochemical processes in the in-cylinder/in-chamber engine and developing efficient combustion technology. However, due to the limited measurement space in the combustion chamber and the strong vibration and flame emission during the operation of the equipment, it is difficult to measure the multi-physical field by the traditional measurement technology. Tomographic laser absorption reconstruction based on laser absorption spectroscopy is a promising diagnostic for mapping the temperature and concentration images because it reduces the need for optical access. Since the inherently ill-posedness of the tomographic inverse problem leads to artifacts, low accuracy, poor robustness, and inability to visualize dynamic flow boundaries in the measured temperature and concentration images, the reconstructed algorithm must contain additional information to promote the presumed solution attributes. To address these problems, we propose a novel algorithm called Gaussian parametric level set method coupled with regularized Landweber to obtain the posterior topological structure of the turbulent reaction flow, which can compensate for the lack of a priori knowledge of the tomographic inverse problem. Numerical validation and algorithm comparison are performed using absorption spectra of water vapor at 7153.748 cm −1 and 7154.353 cm −1 and multi-angle parallel beam arrangement. The proposed approach significantly improves reconstructed quality and has excellent robustness. Even if the sparsely sampled data contains 10% random noise, the root mean square error is still within 0.12 in the temperature range 296–1600 K, approximately 50% of the error of the classical tomography methods. And the standard deviation is about 10% of the reconstructed results of Landweber. The proposed new method can obtain clear dynamic flow field boundaries, eliminate artifacts, improve measurement accuracy and robustness, and its applicability in harsh environments will benefit the supervision of combustion equipment, fuel utilization, pollutant control and optimal design of new combustion systems.

Topology optimization of irregular flow domain by parametric level set method in unstructured mesh

ABSTRACT In this study, we present a novel method for the topology optimization of the irregular flow domain using a parametric level set method (PLSM). Some improvement was applied on the CS-RBFs (radial basis functions with compact support)-based PLSM to make it suitable for nonuniform mesh, expanding the range field of engineering application of the PLSM. The optimization problem is solved by a gradient-based algorithm with Stokes equations as state constraints, and the objective is set to minimize the power dissipation subject to the volume constraint of flow channels. A PLSM is introduced to avoid the direct solving of the Hamilton–Jacobi partial differential equation, which can have the potential to break through the restriction of relying on structured meshes because no finite difference scheme is required. Then, a self-adaption support radius approach is presented to allow the parametric level set to be evolved on the nonuniformed mesh, which can expand the application of the PLSM to more complicated engineering problems with irregular geometric shapes. A volume integration scheme is applied during the design sensitivity analysis to calculate the shape derivatives, allowing the nucleation of new holes. Numerical examples in two and three dimensions are provided to demonstrate the effectiveness of the proposed method.

A parametric level set method for the optimization of composite structures with curvilinear fibers

This paper presents a parametric level set method for the optimization of composite structures with curvilinear fibers. We use iso-contours of a scalar function, also called the level set function, to represent the fibers, and the level set function is constructed by a set of radial basis functions with compact support. The orientation of fiber at any arbitrary point is defined by the orientation of the tangent vector of iso-contour. In order to prevent gaps and overlaps between neighboring fiber tows from arising during the manufacturing of composite structures, a constraint is formulated that the norm of gradient vector of the level set function is equal to 1 almost everywhere. Furthermore, these gradient constraints are aggregated into a single one by using the p-norm method to improve the efficiency of optimization. The compliance minimization problem is considered, and the coefficients of radial basis functions are taken as the design variables. The results of optimizations proved that the proposed method can provide an optimized composite structure with equally spaced curvilinear fibers.

Topology optimization of constrained layer damping plates with frequency- and temperature-dependent viscoelastic core via parametric level set method

Frequency- and temperature-dependent properties of viscoelastic materials have a significant effect on vibration characteristics of constrained layer damping (CLD) structures. In this article, parametrized level set method (PLSM) is extended to solve the topology optimization problem of CLD plates, in which the viscoelastic material with frequency- and temperature-dependent characteristics is employed as damping core. The single/weighted modal loss factors, which are solved by an iterative method, are defined as objective functions, and topology optimization problem is formulated based on the finite element model for CLD plates which are described by combining parametric level set (PLS) functions. The sensitivities of objective functions with respect to expansion coefficients in PLS are calculated by the use of adjoint vector method. As the optimized results with complex constant shear modulus for viscoelastic core indicates, the proposed method is effective for topology optimization of CLD structures and insensitive to initial design. Numerical examples with the consideration of frequency- and temperature-dependent characteristics for “soft” and “hard” viscoelastic core are implemented. Through comparing the optimal results with different models for viscoelastic core, the influences of frequency- and temperature-dependent characteristics for viscoelastic core on natural frequencies, modal loss factors and optimized layouts are analyzed. Similarity indices are defined for distinguishing the difference among optimized layouts and initial design induced by modal strain energy contour. The optimal design dependence on layer thicknesses is further discussed with additional mass of CLD materials unchanged.

A Parametric Level Set Method for Topology Optimization Based on Deep Neural Network

Abstract This paper proposes a new parametric level set method for topology optimization based on deep neural network (DNN). In this method, the fully connected DNN is incorporated into the conventional level set methods to construct an effective approach for structural topology optimization. The implicit function of level set is described by fully connected DNNs. A DNN-based level set optimization method is proposed, where the Hamilton–Jacobi partial differential equations (PDEs) are transformed into parametrized ordinary differential equations (ODEs). The zero-level set of implicit function is updated through updating the weights and biases of networks. The parametrized reinitialization is applied periodically to prevent the implicit function from being too steep or too flat in the vicinity of its zero-level set. The proposed method is implemented in the framework of minimum compliance, which is a well-known benchmark for topology optimization. In practice, designers desire to have multiple design options, where they can choose a better conceptual design base on their design experience. One of the major advantages of the DNN-based level set method is capable to generate diverse and competitive designs with different network architectures. Several numerical examples are presented to verify the effectiveness of the proposed DNN-based level set method.

Parameterized level set method for structural topology optimization based on the Cosserat elasticity

When describing the mechanical behavior of some engineering materials, such as composites, grains, biological materials and cellular solids, the Cosserat continuum theory has more powerful capabilities compared with the classical Cauchy elasticity since an additional local rotation of point and its counterpart (couple stress) are considered in the Cosserat elasticity to represent the material microscale effects. In this paper, a parameterized level set topology optimization method is developed based on the Cosserat elasticity for the minimum compliance problem of the Cosserat solids. The influence of material characteristic length and Cosserat shear modulus on the optimized structure is investigated in detail. It can be found that the microstructural constants in the Cosserat elasticity have a significant impact on the optimized topology configurations. In addition, the minimum feature size and the geometric complexity of the optimized structure can be controlled implicitly by adjusting the parameters of the characteristic length and Cosserat shear modulus easily. Furthermore, the optimized structure obtained by the developed Cosserat elasticity based parameterized level set method will degenerate to the result by using the classical Cauchy elasticity based parameterized level set method when the Cosserat shear modulus approaches zero. A parameterized level set topology optimization method is developed based on the Cosserat elasticity for the optimization of minimum compliance of the structures with micropolar materials. The influence of characteristic length and Cosserat shear modulus of the micropolar material on the optimized structure has been investigated in detail. It can be found that the minimum feature size and the geometric complexity of the optimized structure can be controlled implicitly by adjusting the parameters of the characteristic length and Cosserat shear modulus easily.

Topology Optimization of Nonlinear Material Structures Based on Parameterized Level Set and Substructure Methods

In order to overcome the problems of complicated calculation process and lower computational efficiency of the traditional level set method (LSM), for nonlinear structure topology optimization, a parameterized level set method (PLSM) was introduced. Through interpolation of the initial level set function with the globally supported radial basis function (GSRBF), a nonlinear material structure topology optimization model was established with the interpolation coefficient as the design variable, the minimum strain energy of the structure as the objective function, and the material amount as the constraint condition. The equilibrium equation for the nonlinear material structure was established by finite element analysis, and solved with the iterative method. In addition, the substructure method (i.e. the domain decomposition method) was used to divide the design area into several sub-areas, and the solution to the full degree of freedom equilibrium equation was decomposed into a set of solutions of reduced equilibrium equations and solutions of multiple substructures’ internal displacements, which could reduce the computation cost. Examples show that, this method can ensure the numerical stability, improve the computational efficiency, and obtain the topology optimization configuration with clear boundaries and reasonable structures.

Topology and Shape Optimization of 2-D and 3-D Micro-Architectured Thermoelastic Metamaterials Using a Parametric Level Set Method

2-D and 3-D micro-architectured multiphase thermoelastic metamaterials are designed and analyzed using a parametric level set method for topology optimization and the finite element method. An asymptotic homogenization approach is employed to obtain the effective thermoelastic properties of the multiphase metamaterials. The -constraint multi-objective optimization method is adopted in the formulation. The coefficient of thermal expansion (CTE) and Poisson’s ratio (PR) are chosen as two objective functions, with the CTE optimized and the PR treated as a constraint. The optimization problems are solved by using the method of moving asymptotes. Effective isotropic and anisotropic CTEs and stiffness constants are obtained for the topologically optimized metamaterials with prescribed values of PR under the constraints of specified effective bulk modulus, volume fractions and material symmetry. Two solid materials along with one additional void phase are involved in each of the 2-D and 3-D optimal design examples. The numerical results reveal that the newly proposed approach can integrate shape and topology optimizations and lead to optimal microstructures with distinct topological boundaries. The current method can topologically optimize metamaterials with a positive, negative or zero CTE and a positive, negative or zero Poisson’s ratio.

Functionally Graded Cellular Structure Design Using the Subdomain Level Set Method with Local Volume Constraints

Functional graded cellular structure (FGCS) usually shows superior mechanical behavior with low density and high stiffness. With the development of additive manufacturing, functional graded cellular structure gains its popularity in industries. In this paper, a novel approach for designing functionally graded cellular structure is proposed based on a subdomain parameterized level set method (PLSM) under local volume constraints (LVC). In this method, a subdomain level set function is defined, parameterized and updated on each subdomain independently making the proposed approach much faster and more cost-effective. Additionally, the microstructures on arbitrary two adjacent subdomains can be connected perfectly without any additional constraint. Furthermore, the local volume constraint for each subdomain is applied by virtue of the augmented Lagrange multiplier method. Finally, several numerical examples are given to verify the correctness and effectiveness of the proposed approach in designing the functionally graded cellular structure. From the optimized results, it is also found that the number of local volume constraints has little influence on the convergence speed of the developed approach.

An effective design method for mechanical metamaterials with negative Poisson’s ratios

This paper proposes an effective method for the design of mechanical metamaterials by combining the parametric level set method (PLSM) with the energy-based homogenisation method (EBHM), where the ...

Topology optimization of transient nonlinear heat conduction using an adaptive parameterized level-set method

ABSTRACT In this article, topology optimization of transient nonlinear heat conduction problems is solved by a parameterized level-set method. Nonlinear density-based and continuum shape sensitivity analyses are conducted for topology generation and shape optimization. Meanwhile, the backward time transient adjoint structures are derived. To suppress dependence of the initial guess for the parameterized level-set-based topology optimization, the solid isotropic material with penalization method is adopted to generate the initial topology design. Then, the parameterized level-set method is further performed to obtain the optimal shape. The material volume is strictly controlled by restricting the coefficients of radial basis functions within a dynamic bound, which can avoid incorrect topological results due to topological changes. Finally, three numerical examples are solved with the parameterized level-set method, which demonstrates that the proposed approach can obtain complex topological details with smooth boundaries for transient nonlinear heat conduction problems.

Non-probabilistic Reliability-based Topology Optimization (NRBTO) Scheme for Continuum Structures Based on the parameterized Level-Set method and Interval Mathematics

In this paper, a study on non-probabilistic reliability-based topology optimization (NRBTO) scheme for continuum structures based on the parameterized Level-Set method (PLSM) is conducted, in which the unknown-but-bounded (UBB) uncertainties of material and external loads are taken into account simultaneously. By interpolating the level set function (LSF) with the compactly supported radial basis functions (CSRBFs), the partial differential equation (PDE) is transformed into an ordinary differential equation (ODE). Based on the interval-set model, the displacement constraint is transformed into the non-probabilistic reliability-based scheme and the reliability is evaluated by the optimization feature distance (OFD). Moreover, the interval parametric vertex approach, the concept of shape derivative and the adjoint vector method are employed to obtain the sensitivity between the optimization model and the pseudo time to obtain the evolution velocity field of LSF. By utilizing the optimization criterion (OC) method, the optimization problem can be solved iteratively. To verify the validity and applicability of the proposed NRBTO method, three examples are presented, and numerical results show that taking the UBB uncertainties effects into account during the topology optimization may have a significant influence on the final structural configurations.