Level Set Description Research Articles

A proportional topology optimization method with level-set description and evolutionary strategy

In this work, we propose a proportional topology optimization method with a level-set description and evolutionary strategy (PTO-LSES). A level-set function (LSF) evolution scheme based on the nodal compliance proportion is presented and utilized in the new method. Furthermore, a compliance proportion filtering technique, a regularization scheme, and a material interpolation model with penalty are also used in the proposed method. To determine the effectiveness and superiority of the PTO-LSES method, both the 2D (cantilever beam) and 3D (cantilever and half-MBB beams) numerical examples solving the compliance minimization problem, in contrast to the conventional solid isotropic material with penalization (SIMP) and the original proportional topology optimization (PTO) methods, were tested. In addition, we discuss the impact of compliance proportion filtering and regularization schemes on the new method using 2D half-MBB and cantilever beams. The results showed that compared to the SIMP and PTO methods, the PTO-LSES method offered great advantages via the ability to acquire better objective function (compliance) values and more ideal topology structures with smooth and clear boundaries when subjected to specific conditions. Furthermore, applying the compliance pro-portion filtering technique or regularization scheme to the PTO-LSES method enhanced the performance of the new method in many ways.

Description of random level sets by polynomial chaos expansions

Description of random level sets by polynomial chaos expansions

Local existence of strong solutions to micro–macro models for reactive transport in evolving porous media

AbstractTwo-scale models pose a promising approach in simulating reactive flow and transport in evolving porous media. Classically, homogenised flow and transport equations are solved on the macroscopic scale, while effective parameters are obtained from auxiliary cell problems on possibly evolving reference geometries (micro-scale). Despite their perspective success in rendering lab/field-scale simulations computationally feasible, analytic results regarding the arising two-scale bilaterally coupled system often restrict to simplified models. In this paper, we first derive smooth dependence results concerning the partial coupling from the underlying geometry to macroscopic quantities. Therefore, alterations of the representative fluid domain are described by smooth paths of diffeomorphisms. Exploiting the gained regularity of the effective space- and time-dependent macroscopic coefficients, we present local-in-time existence results for strong solutions to the partially coupled micro–macro system using fixed-point arguments. What is more, we extend our results to the bilaterally coupled diffusive transport model including a level-set description of the evolving geometry.

A VOS based Immersed Boundary-Lattice Boltzmann method for incompressible fluid flows with complex and moving boundaries

A Volume of Solid (VOS) based Immersed Boundary-Lattice Boltzmann Method (IB-LBM) in the framework of the direct forcing based IB-LBM model has been developed in this work to simulate the fluid flow with complex and moving boundaries efficiently. In the present model, the concept of VOS is introduced to achieve the field extension to the solid phase, and a unified Lattice Boltzmann Equation (LBE) has been obtained to describe the fluid flow and the solid body motion consistently. To solve the resulting unified LBE, an efficient direct forcing model has been developed. Compared with the traditional surface based IB model with the direct forcing strategy, in the present work, the dependency of the Lagrangian grid to describe the body profile on the background Cartesian grid is removed by modelling the solid body with a Level-Set function. With such Level-Set description about the body surface, the VOS function can be obtained for the further field extension. With the present IB-LBM algorithm, the motion of the solid body can be enforced effectively without iterations about the forcing term compared with the implicit velocity correction or multiple velocity correction based IB algorithm, and flow penetration, which has been observed in the explicit velocity correction based IB model, can be reduced considerably. To achieve the velocity adjustment in the solid phase, an optimal forcing factor is recommended. With such optimal factor, the unphysical oscillation during force prediction can be well controlled. To verify the performance of the present model, a series of typical benchmarks, including the fluid flow caused by general shaped fixed or moving structures, hydrodynamic characteristics of thin-wall bodies undergoing specified motions and even more complex vortex induced vibrations, are conducted and the good agreements between the present results and the well-validated previous ones confirm the reliability and robustness of the present model.

A reduced-order cutFEM approach to model complex moving sound sources

While traditional finite element based vibro-acoustic analysis of structures is done on stationary structures, in many occasions the structure under investigation is actually moving. To perform accurate finite element analysis on the vibro-acoustic performance of such structures requires the evaluation of a time-varying system, leading to several challenges. Of paramount importance are a strategy to update the mesh to the new configuration and an adequate time integration scheme that does not introduce large numerical artifacts. Additionally, the time-varying nature leads to an increase in computational complexity. In this paper a modeling strategy based on cut finite elements (cutFEM) is introduced that aims to tackle these challenges. The cutFEM approach can be seen as a fictitious domain approach in which the source is moving over a static mesh, cutting through elements where the actual source is located, using a level set description. The key advantage of this approach is that no remeshing and/or mesh morphing is required. It is shown how both implicit and explicit time integration schemes can be used to propagate the solution while the source is moving. Additionally, several model order reduction strategies are compared to speed up the online evaluation of the reduced order model.

Hidden Degrees of Freedom in Implicit Vortex Filaments

This paper presents a new representation of curve dynamics, with applications to vortex filaments in fluid dynamics. Instead of representing these filaments with explicit curve geometry and Lagrangian equations of motion, we represent curves implicitly with a new co-dimensional 2 level set description. Our implicit representation admits several redundant mathematical degrees of freedom in both the configuration and the dynamics of the curves, which can be tailored specifically to improve numerical robustness, in contrast to naive approaches for implicit curve dynamics that suffer from overwhelming numerical stability problems. Furthermore, we note how these hidden degrees of freedom perfectly map to a Clebsch representation in fluid dynamics. Motivated by these observations, we introduce untwisted level set functions and non-swirling dynamics which successfully regularize sources of numerical instability, particularly in the twisting modes around curve filaments. A consequence is a novel simulation method which produces stable dynamics for large numbers of interacting vortex filaments and effortlessly handles topological changes and re-connection events.

A Shape Newton Scheme for Deforming Shells with Application to Capillary Bridges

We present a second order numerical scheme to compute capillary bridges between arbitrary solids by minimizing the total energy of all interfaces. From a theoretical point of view, this approach can be interpreted as the computation of generalized minimal surfaces using a Newton-scheme utilizing the shape Hessian. In particular, we give an explicit representation of the shape Hessian for functionals on shells involving the normal vector without reverting back to a volume formulation or approximating curvature. From an algorithmic perspective, we combine a resolved interface via a triangulated surface for the liquid with a level set description for the constraints stemming from the arbitrary geometry. The actual shape of the capillary bridge is then computed via finite elements provided by the FEniCS environment, minimizing the shape derivative of the total interface energy.

Heat transfer augmentation in microchannel heat sink based on isogeometric topology optimization framework

This paper focuses on a design problem of branching microchannel heat sink based on solving the thermal-fluid problem in isogeometric analysis approach. A brand new topology optimization framework is proposed to determine the distribution of the microchannels, which comprises two coupled computational layers: the upper layer for channel geometry representation and the lower layer for cooling analysis. In the upper layer, several flexible components fitting two Bézier curves as the central line describe the geometry explicitly, which is quite different from the implicit and conventional level-set description ways of topology optimization, fully releasing the deformation ability of the components. By moving, rotating, deforming, bending, overlapping and merging the level-set-based components, the channel geometry gene-rates and is ready to be projected onto the lower layer for the physical field analysis purpose. The lower layer is discretized using a NURBS patch, and NURBS-based isogeometric analysis is adopted to solve the coupled thermal-fluid problem in the structure and calculate the structural sensitivity so as to drive the upper layer to iterate. Comparing with the conventional topology optimization methods, the proposed method has obvious advantages in terms of computational effectiveness and geometry description. A design of multi-layer cold plate is constructed under this framework, aiming at obtaining the best heat dissipation effect under the constraints of volume dissipation and pressure drop. In the final, simulations and experiments are performed to demonstrate these advantages of this framework.

Shape optimization of the time-harmonic response of vibroacoustic devices using cut elements

This article presents a method for generalized shape optimization of time-harmonic vibroacoustic problems. The modeling approach utilizes an immersed boundary cut element method in conjunction with a level set representation of the geometry. The cut element method utilizes a fixed background mesh, a dimensionless contrast parameter and an integration scheme to realize complex geometries and obtain accurate physical solutions to the governing problem. The design parameterization is obtained using a nodal level set description, directly linked to the mathematical design variables, and the gradients of the objective and the constraints are obtained with the discrete adjoint approach. The framework is applied to the optimization of three 2D examples. A study on the effect of initial guess for the proposed optimization procedure is presented on a benchmark example of the design of an acoustic partitioner. Further optimization examples include design of a wave splitter to realize prescribed frequency dependent directivity for emitted acoustic waves and a suspension structure design to improve the performance of a simplified 2D model of a hearing instrument. The results demonstrate that, even though the final topology is strongly dictated by the initial design, modifying the shape allows for a significant improvement of the system behavior. • A generalized shape optimization methodology for 2D time-harmonic vibroacoustic problems using cut elements. • A simple, yet accurate, crisp interface cut element method without the need for stabilization. • A numerically stable level set based shape optimization procedure based on density methods. • Three numerical examples that demonstrate the strengths and weaknesses of the proposed method.

An ODE-driven level-set density method for topology optimization

In this paper, a new topology optimization method is discussed. The basic idea consists of exploring a combination of a density-based method together with a level-set description to form a new optimization frame. Due to the level-set description, solutions show clear and smooth boundaries that are deemed more materially efficient. Additionally, the checkerboarding issue, usually paired with an element-wise description, can be avoided. Thus, the filter scheme to tackle this problem becomes unnecessary, and the design space can be further exploited. The ingredient of material interpolation with penalty is utilized here, and its merit lies in making update information (i.e., sensitivity) more distinguished and in turn driving the optimization process to converge into solid–void solutions stably. An ordinary differential equation (ODE) established from optimal criteria builds the relationship between the level-set description and update information, and the structural updating procedure can be efficiently performed by solving the ODE. A regularization scheme for the level-set is introduced to enhance topological variation ability and address topological evolution defects, which helps deliver reasonable and well-posed topological configurations. The regularization strategy is a simple linear scaling manner that proves to be effective and efficient. This paper investigates three classes of optimization problems: compliance minimization, eigenfrequency maximization, and thermal conduction optimization. To validate the proposed method, both 2D and 3D benchmark examples in comparison with the widely accepted Solid Isotropic Material with Penalization (SIMP) method are tested. By contrast, the solutions resulting from the proposed method show advantageous structural representations and better objective function values under specified conditions. In addition, several other numerical examples considering model parameter influences and some extensions are discussed to systematically demonstrate the proposed method’s characteristics. Finally, the MATLAB codes concerning the above-mentioned three classes of optimization problems are shared for educational purposes. The codes are compact and finely structured with only 58, 62, and 53 lines for 2D cases, and 105, 108, and 80 lines for 3D cases of minimum compliance, maximum eigenfrequency, and minimum thermal compliance problems, respectively.

Topology optimization of high frequency vibration problems using the EFEM-based approach

The energy finite element method (EFEM) provides researchers with an efficient tool to analyze high-frequency vibrating solid structures with less calculation and clear distribution of the energy density. However, the corresponding applications in structure optimization mainly focus on modifying the size and material properties, and it leaves a scientific gap that the topology flexibility is not addressed enough in the optimization. Therefore, this work aims at establishing an explicit level-set based topology optimization framework for the energy finite element method. A series of basic technical aspects, including the explicit level set description method, customized finite element mesh, mathematical model, and sensitivity analysis, are presented. With these basic studies, an original EFEM-based topology optimization framework for thin-walled structures is established for the first time. It is a basic work to conduct topology optimization in terms of energy. Additional applications in curved surfaces, spatial structures, and compound materials can also be developed only by modifying the description and EFEM module. Finally, the proposed optimization is applied to classic stiffened plates. Further analyses indicate that the proposed EFEM-based topology optimization can improve the dynamic performance of high-frequency vibrating structures by 20%–80%.

An interface-enriched generalized finite element method for level set-based topology optimization

During design optimization, a smooth description of the geometry is important, especially for problems that are sensitive to the way interfaces are resolved, e.g., wave propagation or fluid-structure interaction. A level set description of the boundary, when combined with an enriched finite element formulation, offers a smoother description of the design than traditional density-based methods. However, existing enriched methods have drawbacks, including ill-conditioning and difficulties in prescribing essential boundary conditions. In this work, we introduce a new enriched topology optimization methodology that overcomes the aforementioned drawbacks; boundaries are resolved accurately by means of the Interface-enriched Generalized Finite Element Method (IGFEM), coupled to a level set function constructed by radial basis functions. The enriched method used in this new approach to topology optimization has the same level of accuracy in the analysis as the standard finite element method with matching meshes, but without the need for remeshing. We derive the analytical sensitivities and we discuss the behavior of the optimization process in detail. We establish that IGFEM-based level set topology optimization generates correct topologies for well-known compliance minimization problems.

A novel highly efficient Lagrangian model for massively multidomain simulation applied to microstructural evolutions

In this paper, a new method for the simulation of evolving multi-domains problems will be presented. The method is inspired by the front-tracking approaches where only the interfaces between domains are discretized. The presented method maintains the discretization of the interior of the domains using an evolving triangular mesh (valid for a Finite Element (FE) study) and treats the topological events such as the disappearance of domains or the creation of interfaces by means of selective local remeshing operations. Geometric properties are only computed on the interfaces using local spline reconstructions and interfaces are moved using a Lagrangian approach ensuring at all times the validity of the mesh. Accuracy and computational cost of this method will be evaluated in the context of microstructural evolutions, specifically for grain growth mechanism (GG) and it will be compared to a more classical front capturing approach based on a Level-Set (LS) description of grain interfaces which have been proved to be a powerful method for this kind of simulations.

Kostant–Toda lattices and the universal centralizer

To each complex semisimple Lie algebra g decorated with appropriate data, one may associate two completely integrable systems. One is the well-studied Kostant–Toda lattice, while the second is an integrable system defined on the universal centralizer Zg of g. These systems are similar in that each exploits and closely reflects the invariant theory of g, as developed by Chevalley, Kostant, and others. One also has Kostant’s description of level sets in the Kostant–Toda lattice, which turns out to suggest deeper similarities between the two integrable systems in question. Some more recent works allude to a strong connection between the two systems, e.g. Bezrukavnikov et al. (2005), Teleman (2014, 2018).We study relationships between the two aforementioned integrable systems, partly to understand and contextualize the similarities mentioned above. Our main result is a canonical open embedding of a flow-invariant open dense subset of the Kostant–Toda lattice into Zg. Secondary results include some qualitative features of the integrable system on Zg.

Stability and error analysis for a diffuse interface approach to an advection\u2013diffusion equation on a moving surface

In this paper we analyze a fully discrete numerical scheme for solving a parabolic PDE on a moving surface. The method is based on a diffuse interface approach that involves a level set description of the moving surface. Under suitable conditions on the spatial grid size, the time step and the interface width we obtain stability and error bounds with respect to natural norms. Furthermore, we present test calculations that confirm our analysis.

EXtended Hybridizable Discontinuous Galerkin with Heaviside Enrichment for Heat Bimaterial Problems

A novel strategy for the hybridizable discontinuous Galerkin (HDG) solution of heat bimaterial problems is proposed. It is based on eXtended finite element philosophy, together with a level set description of interfaces. Heaviside enrichment on cut elements and cut faces is used to represent discontinuities across the interface. A suitable weak form for the HDG local problem on cut elements is derived, accounting for the discontinuous enriched approximation, and weakly imposing continuity or jump conditions over the material interface. The computational mesh is not required to fit the interface, simplifying and reducing the cost of mesh generation and, in particular, avoiding continuous remeshing for evolving interfaces. Numerical experiments demonstrate that X-HDG keeps the accuracy of standard HDG methods in terms of optimal convergence and superconvergence.

Improvement of 3D mean field models for capillarity-driven grain growth based on full field simulations

In the present study, mean field models of grain growth (Hillert and Burke–Turnbull models) are compared with 3D full field simulations considering an isotropic grain boundary energy and mobility and under the absence of second-phase particles. The present 3D full field simulations are based on a level set description of the grain interfaces within a finite element framework. The digital initial microstructures are generated using a coupled “Voronoi–Laguerre/dense sphere packing” algorithm. Based on full field simulation results, new formulations of Burke–Turnbull and Hillert models are proposed. In contrast with classical formulations, the new ones account for the possible heterogeneity of the initial grain size distribution.

Analytical sensitivity analysis using the extended finite element method in shape optimization of bimaterial structures

The present work investigates the shape optimization of bimaterial structures. The problem is formulated using a level set description of the geometry and the extended finite element method (XFEM) to enable an easy treatment of complex geometries. A key issue comes from the sensitivity analysis of the structural responses with respect to the design parameters ruling the boundaries. Even if the approach does not imply any mesh modification, the study shows that shape modifications lead to difficulties when the perturbation of the level sets modifies the set of extended finite elements. To circumvent the problem, an analytical sensitivity analysis of the structural system is developed. Differences between the sensitivity analysis using FEM or XFEM are put in evidence. To conduct the sensitivity analysis, an efficient approach to evaluate the so-called velocity field is developed within the XFEM domain. The proposed approach determines a continuous velocity field in a boundary layer around the zero level set using a local finite element approximation. The analytical sensitivity analysis is validated against the finite differences and a semianalytical approach. Finally our shape optimization tool for bimaterial structures is illustrated by revisiting the classical problem of the shape of soft and stiff inclusions in plates. Copyright c © 2010 John Wiley & Sons, Ltd.

New finite element developments for the full field modeling of microstructural evolutions using the level-set method

Recently a new numerical model devoted to the full field modeling of microstructural evolutions at the polycrystal scale has been proposed and validated [1]. The latter is based on a level set description of interfaces in a finite element framework. Firstly introduced to model 2D and 3D primary recrystallization with nucleaction [1,2], it has then been extended to consider the grain growth stage [3,4]. The ability of this approach to model the Zener pinning phenomenon without any assumption concerning the shape of second phase particles was also demonstrated [5]. This model has nevertheless an elevated computational cost and requires many numerical parameters whose calibration is not straightforward. In the present paper, some major improvements of the model which address these two points are discussed. A comparative study is also provided in order to illustrate the gains achieved in terms of computational efficiency and robustness.

Advances in Level-Set Modeling of Recrystallization at the Polycrystal Scale - Development of the Digi-<i>μ</i> Software

The mechanical and thermal properties of metallic materials are strongly related to theirmicrostructure. An accurate and quantitative prediction of microstructural evolutions is then crucialwhen it comes to optimize the forming process. Recently a new full field approach, based on a Level-Set (LS) description of interfaces in a finite element (FE) context has been introduced to model 2D and3D primary recrystallization (ReX), including the nucleation stage [1, 2], and has been extended to takeinto account the grain growth (GG) stage [3, 4]. The ability of this approach to model also the Zenerpinning (ZP) phenomenon without any assumption concerning the shape of second phase particleswas also demonstrated [5]. Moreover, recent developments have also illustrated the capability of thisapproach to take into account the characteristics of twin interfaces during grain boundary motion [6,7]. Current work concerns also the improvement of the numerical cost of this new approach [8]. Allthese developments are necessary to account for the microstructural complexity of ReX phenomenon.