Multi-material Problems Research Articles

Isogeometric proportional topology optimization (IGA-PTO) for multi-material problems

This study addresses the multi-material optimization problem by the effective and robust isogeometric proportional topology optimization (IGA-PTO) approach. Non-uniform rational B-spline basis functions are used for descriptions of geometry, displacement field, and density fields of material phases. The alternating active-phase algorithm is employed to convert the optimization problem with multiple constraints into a series of sub-problems of single constraint. Several examples are exhibited, and intensive comparisons with the gradient-based optimality criteria (OC) algorithm are presented to highlight the superior performance of the proposed PTO algorithm. Finally, the optimized topologies are prototyped using 3D printing technology to evidence the manufacturing feasibility.

Mixed displacement–pressure formulations and suitable finite elements for multimaterial problems with compressible and incompressible models

Multimaterial problems in linear and nonlinear elasticity are some of the least explored using mixed finite element formulations with higher-order elements. The fundamental issue in adapting the mixed displacement–pressure formulations with linear and higher-order continuous elements for the pressure field is their inability to capture pressure and stress jumps across material interfaces. In this paper, for the first time in literature, we perform comprehensive studies of multimaterial problems in elasticity consisting of compressible and incompressible material models using the mixed displacement–pressure formulation to assess the performance of different element types in accurately resolving pressure fields within the domains and pressure jumps across material interfaces. In particular, inf–sup stable displacement–pressure combinations with element-wise discontinuous pressure for triangular and tetrahedral elements are considered and their performance is assessed along with the Q1/P0 element and Taylor–Hood elements using several numerical examples. The results show that Taylor–Hood elements fail to capture the stress jumps due to the continuity of DOFs across elements, the Crouzeix–Raviart (P2b/P1dc) element yields substantially poor pressure fields despite a significant increase in pressure degrees of freedom and that the P3/P1dc element produces superior quality results fields when compared with the P2b/P1dc element.

Stress-driven design of incompressible multi-materials under frequency constraints

This paper presents an efficient and comprehensive numerical approach to address topology optimization challenges within the multi-material framework. The focus encompasses several key aspects: solving various stress-related problems from compressible to incompressible materials, broadening the scope to encompass multi-material systems, adopting a flexible methodology suitable for various elements (triangular, quadrilateral, and polygonal), and expanding to incorporate frequency constraints within the stress-driven system. The core idea to accommodate a broad range of materials, from compressible to nearly incompressible, is to implement a specialized polytopal composite finite element (PCE) technique to mitigate volumetric locking issues often prevalent in nearly incompressible materials. Then, to effectively solve the stress-related multi-material system, a well-known P-norm function integrated with the Moved and Regularized Heaviside function (MRHF) is employed, capitalizing on the correlation between topological phases and material allocation to handle multi-material problems effectively. Additionally, to deal with frequency-related multi-material problems, an extended technique addressing localized mode issues based on the relationship between representative solid and void is also employed. Several numerical examples are tested to validate the method’s efficiency and reliability within a multi-material framework, considering a diverse range of materials from compressible to nearly incompressible.

Various challenges in numerical simulation of three-dimensional multi-physics and multi-material problems under extreme 条件

Various challenges in numerical simulation of three-dimensional multi-physics and multi-material problems under extreme 条件

A polygonal topology optimization method based on the alternating active-phase algorithm

<abstract> <p>We propose a polygonal topology optimization method combined with the alternating active-phase algorithm to address the multi-material problems. During the process of topology optimization, the polygonal elements generated by signed distance functions are utilized to discretize the structural design domain. The volume fraction of each material is considered as a design variable and mapped to its corresponding element variable through a filtering matrix. This method is used to solve a multi-material structural topology optimization problem of minimizing compliance, in which a descriptive model is established by using the alternating active-phase algorithm and the solid isotropic microstructure with penalty theory. This method can accomplish the topology optimization of multi-material structures with complex curve boundaries, eliminate the phenomena of checkerboard patterns and a one-node connection, and avoid sensitivity filtering. In addition, this method possesses fine numerical stability and high calculation accuracy compared to the topology optimization methods that use quadrilateral elements or triangle elements. The effectiveness and feasibility of this method are demonstrated through several commonly used and representative numerical examples.</p> </abstract>

Proportional Topology Optimization algorithm with virtual elements for multi-material problems considering mass and cost constraints

This paper presents an extension of the Proportional Topology Optimization (PTO) with virtual elements for multi-material problems with mass and cost constraints. In particular, the linear virtual element method (VEM) is constructed on unstructured polygonal meshes. The linear VEM is desirable in the sense that numerical integration is not explicitly required, significantly reducing the computational effort. Furthermore, the unstructured polygonal mesh naturally eliminates the issue of one-node connections encountered by the usual quadrilateral mesh. A feature of PTO is that it does not require sensitivity information, i.e., the derivative of the objective function with respect to design variables. Instead, the amount of material distributed into each element is determined proportionally to the contribution of that element to the objective function. For multi-material problems, the Ordered Solid Isotropic Material with Penalization (Ordered SIMP) technique is integrated into the PTO framework. Compared to other techniques for problems that involve multiple materials, Ordered SIMP has the advantage that computational cost does not depend on the number of materials. Furthermore, for the first time, the PTO approach is extended to consider two types of constraints: mass and cost simultaneously. The feasibility and efficiency of the proposed method are demonstrated via several benchmark examples and comparisons with the existing approach.

Enriched finite element approach for modeling discontinuous electric field in multi-material problems

Enriched finite element approach for modeling discontinuous electric field in multi-material problems

High order conservative Lagrangian scheme for three-temperature radiation hydrodynamics

The three-temperature (3-T) radiation hydrodynamics (RH) equations are widely used in modeling various optically thick high-energy-density-physics environments, such as those in astrophysics and inertial confinement fusion (ICF). In this paper, we will discuss the methodology to construct a high order conservative Lagrangian scheme solving 3-T RH equations. Specifically, the three new energy variables are defined first, in the form of which the three energy equations of the 3-T RH equations are rewritten. The main advantage of this formulation is that it facilitates the design of a scheme with both conservative property and arbitrary high order accuracy. Starting from one dimension and based on the multi-resolution WENO reconstruction and the strong stability preserving (SSP) high order time discretizations, taken as an example, we design a third order conservative Lagrangian scheme both in space and time. To determine the numerical flux for the conservative advection terms in the 3-T RH equations, we propose a HLLC numerical flux which is derived from the divergence theorem rigorously and is suitable for multi-material problems with the ideal-gas equations of state. After that, we discuss how to design a class of high order positivity-preserving explicit Lagrangian schemes to solve the 3-T RH equations which only contain the conservative advection terms in space. Preliminary extension to the two dimensional case is also considered. Finally, various numerical tests are given to verify the desired properties of the high order Lagrangian schemes such as high order accuracy, non-oscillation, conservation and adaptation to multi-material problems.

On the design of stable, consistent, and conservative high-order methods for multi-material hydrodynamics

Obtaining stable and high-order numerical solutions for multi-material hydrodynamics is an open challenge. Although slope limiters are widely used to maintain monotonicity near discontinuities, typical limiting procedures violate closure laws at the discrete level when applied to multi-material hydrodynamics equations. Due to this, the high-order expansions of quantities related by the closure laws are no longer consistent. The commonly observed symptom of this consistency-violation is that the numerical method fails to maintain constant pressure and velocity across material interfaces. This leads to sub-optimal convergence rates for smooth multi-material problems as well. Specialized limiting procedures that satisfy consistency while maintaining conservation need to be developed for such equations. A novel procedure that re-instates consistency into slope-limited high-order discretizations applied to the multi-material hydrodynamics equations is presented here. Using simple examples, it is demonstrated that the presented method satisfies closure laws at the discrete level, while maintaining conservative properties of the high-order method. Furthermore, this procedure involves a projection step which relies on the compact basis of the underlying spatial discretization, i.e. for discontinuous schemes (viz. DG and FV) the projection is local, and does not involve global matrix solves. Comparisons with conventional approaches emphasizes the necessity of the consistent closure-law preserving limiting approach, in order to maintain design order of accuracy for smooth multi-material problems.

Solving multi-material problems in solid mechanics using physics-informed neural networks based on domain decomposition technology

Physics-informed neural networks (PINNs) are widely used in the field of solid mechanics. Currently, PINNs are mainly used to solve problems involving single homogeneous materials. However, they have limited ability to handle the discontinuities that arise from multi-material, and they lack the capability to rigorously express complex material contact models. We propose a method for solving multi-material problems in solid mechanics using physics-informed neural networks. Inspired by domain decomposition technology, the calculation domain is divided according to the geometric distribution of materials, with different subnetworks applied to represent field variables. This study explains how the invariant momentum balance, kinematic relations, and different constitutive relations controlled by the material properties are incorporated into the subnetworks, and use additional regular terms to describe the contact relations between materials. Various test cases ranging from two-dimensional plane strain problems to three-dimensional stretching problems are solved using the proposed method. We introduce the concept of parameter sharing in multi-task learning (MTL) and incorporate it in the proposed method, which yields additional degrees of freedom in choosing the sharing structure and sharing mode. Compared with common physics-informed neural network algorithms, which are based on fully independent parameters, we develop a network structure with partial sharing structure and all-sharing mode that achieves higher accuracy when solving the example problems.

Boolean finite cell method for multi-material problems including local enrichment of the Ansatz space

AbstractThe Finite Cell Method (FCM) allows for an efficient and accurate simulation of complex geometries by utilizing an unfitted discretization based on rectangular elements equipped with higher-order shape functions. Since the mesh is not aligned to the geometric features, cut elements arise that are intersected by domain boundaries or internal material interfaces. Hence, for an accurate simulation of multi-material problems, several challenges have to be solved to handle cut elements. On the one hand, special integration schemes have to be used for computing the discontinuous integrands and on the other hand, the weak discontinuity of the displacement field along the material interfaces has to be captured accurately. While for the first issue, a space-tree decomposition is often employed, the latter issue can be solved by utilizing a local enrichment approach, adopted from the extended finite element method. In our contribution, a novel integration scheme for multi-material problems is introduced that, based on the B-FCM formulation for porous media, originally proposed by Abedian and Düster (Comput Mech 59(5):877–886, 2017), extends the standard space-tree decomposition by Boolean operations yielding a significantly reduced computational effort. The proposed multi-material B-FCM approach is combined with the local enrichment technique and tested for several problems involving material interfaces in 2D and 3D. The results show that the number of integration points and the computational time can be reduced by a significant amount, while maintaining the same accuracy as the standard FCM.

A thermal equilibrium approach to modelling multiple solid–fluid interactions with phase transitions, with application to cavitation

This paper presents a new approach to model phase transitions in multi-material problems based on the assumption of thermal equilibrium. The method is used to construct an equation of state for water which captures the liquid–vapour phase transition and is suitable for simulating cavitating flows. A key advantage of this approach is that the physics of the phase transition is entirely encoded within the equation of state. Complex multi-physics models can therefore be augmented to include the physics of phase transitions without further complicating the partial evolution equations. This is substantiated by integrating the model within a diffuse interface scheme for coupled solid–fluid dynamics, and implemented using a structured adaptive mesh refinement (SAMR) strategy. The combined model facilitates the fully-coupled simulation of complex multi-material, multi-physics problems featuring elastoplastic solids, gases, cavitating liquids and reactive materials. Evaluation of the model is shown to add little overhead – for parts of the domain without cavitation the equations reduce to those of the underpinning model with simple stiffened gas equation of state for liquid components; for parts of the domain where cavitation does occur, it requires a non-linear relaxation with only one degree of freedom. The method is applied to several strenuous test cases for cavitating flows and shown to be in good agreement with previously published works.

XIGA: An eXtended IsoGeometric analysis approach for multi-material problems

Multi-material problems often exhibit complex geometries along with physical responses presenting large spatial gradients or discontinuities. In these cases, providing high-quality body-fitted finite element analysis meshes and obtaining accurate solutions remain challenging. Immersed boundary techniques provide elegant solutions for such problems. Enrichment methods alleviate the need for generating conforming analysis grids by capturing discontinuities within mesh elements. Additionally, increased accuracy of physical responses and geometry description can be achieved with higher-order approximation bases. In particular, using B-splines has become popular with the development of IsoGeometric Analysis. In this work, an eXtended IsoGeometric Analysis (XIGA) approach is proposed for multi-material problems. The computational domain geometry is described implicitly by level set functions. A novel generalized Heaviside enrichment strategy is employed to accommodate an arbitrary number of materials without artificially stiffening the physical response. Higher-order B-spline functions are used for both geometry representation and analysis. Boundary and interface conditions are enforced weakly via Nitsche’s method, and a new face-oriented ghost stabilization methodology is used to mitigate numerical instabilities arising from small material integration subdomains. Two- and three-dimensional heat transfer and elasticity problems are solved to validate the approach. Numerical studies provide insight into the ability to handle multiple materials considering sharp-edged and curved interfaces, as well as the impact of higher-order bases and stabilization on the solution accuracy and conditioning.

Adaptive isogeometric multi-material topology optimization based on suitably graded truncated hierarchical B-spline

A novel adaptive multi-material topology optimization (TO) is presented in this work based on suitably graded truncated hierarchical B-spline in the framework of isogeometric analysis (IGA). The isogeometric mesh is locally refined and coarsened during the topology process to achieve the analysis and design meshes for multi-material design problems. Adaptive marking strategy is established based on the summation of density variation indicators of all material phases, which plays a critical role in obtaining elements with different phases to be refined and coarsened. Besides, the graded constraint is imposed on the hierarchical isogeometric mesh to improve the numerical accuracy of multi-material problems solved by adaptive IGA and ensure the boundary quality of the converged multi-material structures. Through an alternating active-phase algorithm in a hierarchy form, the proposed isogeometric multi-material TO model can be divided into finite two-phase adaptive isogeometric optimization subproblems, of which the sensitivity analysis is derived in terms of an adaptively adjusted filter. Compared with the current isogeometric multi-resolution multi-material method, the proposed adaptive one can obtain optimized multi-material designs with much-improved efficiency. Several numerical examples are applied to verify the robustness and effectiveness of the proposed approach.

A flux-enriched Godunov method for multi-material problems with interface slide and void opening

This work outlines a new three-dimensional diffuse interface finite volume method for the simulation of multiple solid and fluid components featuring large deformations, sliding and void opening. This is achieved by extending an existing reduced-equation diffuse interface method by means of a number of novel flux-modifiers and interface seeding routines that enable the application of different material boundary conditions. The method allows for slip boundary conditions across solid interfaces, material-void interaction, and interface separation. The method is designed to be straightforward to implement, inexpensive and highly parallelisable. This makes it suitable for use in large, multi-dimensional simulations that feature many complex materials and physical processes interacting over multiple levels of adaptive mesh refinement. Furthermore, the method allows for the generation of new interfaces in a conservative fashion and therefore naturally facilitates the simulation of high-strain rate fracture. Hence, the model is augmented to include ductile damage to allow for validation of the method against demanding physical experiments. The method is shown to give excellent agreement with both experiment and existing Eulerian interface tracking algorithms that employ sharp interface methods.

A stabilised displacement–volumetric strain formulation for nearly incompressible and anisotropic materials

The simulation of structural problems involving the deformations of volumetric bodies is of paramount importance in many areas of engineering. Although the use of tetrahedral elements is extremely appealing, tetrahedral discretisations are generally known as very stiff and are hence often avoided in typical simulation workflows.The development of mixed displacement–pressure approaches has allowed to effectively overcome this problem leading to a class of locking-free elements which can effectively compete with hexahedral discretisations while retaining obvious advantages in the mesh generation step. Despite such advantages the adoption of the technology within commercial codes is not yet pervasive.This can be attributed to two different reasons: the difficulty in making use of standard constitutive libraries and the implied continuity of the pressure, which makes the application of the method questionable in the context of multi-material problems. Current paper proposes the adoption of the volumetric strain instead of the pressure as a nodal value. Such choice leads to the definition of a modified strain making the use of standard strain-driven constitutive laws straightforward. At the same time, the continuity of the volumetric strain across multimaterial interfaces can be understood as a sort of kinematic constraint (stresses can still remain discontinuous across material interfaces). The new element also opens the door to the use of anisotropic constitutive laws, which are typically problematic in the context of mixed elements.

High order conservative Lagrangian schemes for one-dimensional radiation hydrodynamics equations in the equilibrium-diffusion limit

Radiation hydrodynamics (RH) describes the interaction between matter and radiation which affects the thermodynamic states and the dynamic flow characteristics of the matter-radiation system. Its application areas are mainly in high-temperature hydrodynamics, including gaseous stars in astrophysics, combustion phenomena, reentry vehicles fusion physics and inertial confinement fusion (ICF). Solving the radiation hydrodynamics equations (RHE), even in the equilibrium-diffusion limit, is a difficult task. In this paper, we will discuss the methodology to construct fully explicit and implicit-explicit (IMEX) high order Lagrangian schemes solving one dimensional RHE in the equilibrium-diffusion limit respectively, which can be used to simulate multi-material problems with the coupling of radiation and hydrodynamics. The schemes are based on the HLLC numerical flux, the essentially non-oscillatory (ENO) reconstruction for the advection term, ENO reconstruction or high order central reconstruction and interpolation for the radiation diffusion term, the Newton iteration method (for the IMEX scheme), and the strong stability preserving (SSP) high order time discretizations. The schemes can maintain conservation and uniformly high order accuracy both in space and time. The issue of positivity-preserving for the high order explicit Lagrangian scheme is also discussed. Various numerical tests for the high order Lagrangian schemes are provided to demonstrate the desired properties of the schemes such as high order accuracy, non-oscillation, and positivity-preserving.

3D Cell-centered hydrodynamics with subscale closure model and multi-material remap

We extend a higher-order finite volume cell-centered hydrodynamic (CCH) formulation to include an interface-aware subscale closure model and a multi-material remap for simulating 3D compressible hydrodynamic problems within an arbitrary Lagrangian-Eulerian (ALE) framework. This CCH formulation involves a multidirectional approximate Riemann solution using quadratic polynomial reconstructions of the stress tensor and the velocity. At the subscale level, we determine pair-wise material interactions by solving a distinct approximate Riemann problem at the common interface, using the volume of fluids (VOF) method to find the interface. Material interactions are constrained to ensure smooth pressure equilibration among materials. The accuracy and robustness of the ALE method is demonstrated by simulating a suite of 3D Cartesian multi-material problems covering both gas and solid dynamics, where each test case has two or more materials.

Efficient multi-material continuum topology optimization considering hyperelasticity: Achieving local feature control through regional constraints

We introduce a general and efficient multi-material topology optimization framework considering hyperelasticity with many local constraints, which enables flexible control of the design’s local features. The proposed framework can effectively distribute multiple candidate materials described by distinct constitutive models according to their respective nonlinear behaviors, and efficiently handle a flexible setting of volume constraints, either global or local. To ensure computational efficiency of the proposed framework, we present a virtual element-based formulation in conjunction with a tailored adaptive refinement and coarsening scheme for multi-material problems, and we adopt the ZPR scheme to update the design variables associated with each constraint in parallel. Three design examples are presented, demonstrating the efficiency and effectiveness of the proposed framework in distributing multiple candidate materials with distinct nonlinear elastic behaviors and handling both global and many (e.g., 1024) local constraints. We envision that the proposed framework enables unique computational capabilities for designing next-generation composite metamaterials and structures with nonlinear behaviors and multi-functionalities.

A unified Eulerian framework for multimaterial continuum mechanics

A framework for simulating the interactions between multiple different continua is presented. Each constituent material is governed by the same set of equations, differing only in terms of their equations of state and strain dissipation functions. The interfaces between any combination of fluids, solids, and vacuum are handled by a new Riemann Ghost Fluid Method, which is agnostic to the type of material on either side (depending only on the desired boundary conditions).The Godunov-Peshkov-Romenski (GPR) model is used for modelling the continua (having recently been used to solve a range of problems involving Newtonian and non-Newtonian fluids, and elastic and elastoplastic solids), and this study represents a novel approach for handling multimaterial problems under this model.The resulting framework is simple, yet capable of accurately reproducing a wide range of different physical scenarios. It is demonstrated here to accurately reproduce analytical results for known Riemann problems, and to produce expected results in other cases, including some featuring heat conduction across interfaces, and impact-induced deformation and detonation of combustible materials. The framework thus has the potential to streamline development of simulation software for scenarios involving multiple materials and phases of matter, by reducing the number of different systems of equations that require solvers, and cutting down on the amount of theoretical work required to deal with the interfaces between materials.