2D Examples Research Articles

A fourth-order compact time-splitting method for the Dirac equation with time-dependent potentials

In this paper, we present an approach to deal with the dynamics of the Dirac equation with time-dependent electromagnetic potentials using the fourth-order compact time-splitting method (S4c). To this purpose, the time-ordering technique for time-dependent Hamiltonians is introduced, so that the influence of the time-dependence could be limited to certain steps which are easy to treat. Actually, in the case of the Dirac equation, it turns out that only those steps involving potentials need to be amended, and the scheme remains efficient, accurate, as well as easy to implement. Numerical examples in 1D and 2D are given to validate the scheme.

Convergence and Optimality Analysis of Low-Dimensional Generative Adversarial Networks Using Error Function Integrals

Due to their success at synthesising highly realistic images, many claims have been made about optimality and convergence in generative adversarial networks (GANs). But what of vanishing gradients, saturation, and other numerical problems noted by AI practitioners? Attempts to explain these phenomena have so far been based on purely empirical studies or differential equations, valid only in the limit. We take a fresh look at these questions using explicit, low-dimensional models. We revisit the well known optimal discriminator result and, by construction of a counterexample, show that it is not valid in the case of practical interest: when the dimension of the latent variable is less than that of the data: dim(z) < dim(x). To examine convergence issues, we consider a 1-D least squares (LS) GAN with exponentially distributed data, a Rayleigh distributed latent variable, a square law generator and a discriminator of the form D(x) = (1 + erf(x))=2 where erf is the error function. We obtain explicit representations of the cost (or loss) function and its derivatives. The representation is exact down to the evaluation of a well-behaved 1-D integral. We present analytical numerical examples of 2D and 4D parameter trajectories for gradient-based minimax optimisation. Although the cost function has no saddle points, it generally has a minimum, maximum and plateaux areas. The gradient algorithms typically converge to a plateau, where the gradients vanish and the cost function saturates. This is an undesirable setting with no implications of optimality for either the generator or discriminator. The analytical method is compared with stochastic gradient optimisation and proven to be a very accurate predictor of the latter’s performance. The quasi-deterministic framework we develop is a powerful analytical tool for understanding convergence behaviour of low-dimensional GANs based on least-squares cost criteria.

Error analysis of a conforming and locking-free four-field formulation for the stationary Biot’s model

We present ana priorianda posteriorierror analysis of a conforming finite element method for a four-field formulation of the steady-state Biot’s consolidation model. For thea priorierror analysis we provide suitable hypotheses on the corresponding finite dimensional subspaces ensuring that the associated Galerkin scheme is well-posed. We show that a suitable choice of subspaces is given by the Raviart–Thomas elements of orderk≥ 0 for the fluid flux, discontinuous polynomials of degreekfor the fluid pressure, and any stable pair of Stokes elements for the solid displacements and total pressure. Next, we develop a reliable and efficient residual-baseda posteriorierror estimator. Both the reliability and efficiency estimates are shown to be independent of the modulus of dilatation. Numerical examples in 2D and 3D verify our analysis and illustrate the performance of the proposeda posteriorierror indicator.

3D Virtual Elements for Elastodynamic Problems

AbstractA virtual element framework for nonlinear elastodynamics is outlined within this work. The virtual element method (VEM) can be considered as an extension of the classical finite element method. While the finite element method (FEM) is restricted to the usage of regular shaped elements, VEM allows to use non‐convex shaped elements for the spatial discretization [1]. It has been applied to various engineering problems in elasticity and other areas, such as plasticity or fracture mechanics as outlined in [3, 4]. This work deals with the extension of VEM to dynamic problems. Low‐order ansatz functions in two and three dimensions, with elements being arbitrary shaped, are used in this contribution. The formulations considered in this framework are based on minimization of energy, where a pseudo potential is used for the dynamic behavior. While the stiffness‐matrix needs a suitable stabilization, the mass‐matrix can be calculated fully through the projection part. For the implicit time integration, Newmark‐Method is used. To show the performance of the method, various numerical examples in 2D and 3D are presented.

A Scaled Boundary Approach for Inelasticity of Fibre‐Reinforced Composites

AbstractAs versatile materials with elevated mechanical properties, short fibre reinforced composites (SFRCs) are currently used in many applications. To enhance the mechanical properties, e.g. the ductility, or to reduce crack propagation, short fibres are added to a matrix material, such as polymers or concrete. In the case of concrete, for example, the composite might contain additional aggregates as well. Although the addition of fibres improves the mechanical properties in general, it is well‐known that many matrix materials exhibit creep deformations under constant long‐term loads.The contribution at hand discusses a scaled boundary approach to model the inelastic behaviour of SFRCs. To allow for an efficient and automated mesh generation, the scaled boundary finite element method (SBFEM) is used in conjunction with the quadtree/octree decomposition algorithm. Within this framework, the fibres are modelled in a discrete manner, and the mesh of the matrix is generated independently of the fibres. Applying a novel embedding method, the meshes of the matrix and the fibres are combined, which results in a compatible mesh without requiring additional constraints. Furthermore, it is straightforward to combine the discussed techniques with image‐based mesh generation, e.g. based on computed tomography scans. The current contribution presents various 2D and 3D examples illustrating the efficiency and adaptability of the proposed framework.In addition, we present a methodology to incorporate non‐linear constitutive models into the SBFEM. Separate constitutive equations are formulated for the matrix, aggregates, and fibres. Whereas the mechanical behaviour of the aggregates and the fibres can be modelled within the framework of elasticity, one must make use of non‐linear approaches to account for the rate‐dependent inelastic deformation of the matrix under multiaxial stress and deformation states. Overall, this method allows for the simulation of creep in SFRCs, while considering the complex microstructure of these materials in detail. This provides the basis for a realistic estimation of the effect of long‐term loading on SFRC components.

Stress Relaxation and Sensitivity Weight for Bi-Directional Evolutionary Structural Optimization to Improve the Computational Efficiency and Stabilization on Stress-Based Topology Optimization

Stress-based topology optimization is one of the most concerns of structural optimization and receives much attention in a wide range of engineering designs. To solve the inherent issues of stress-based topology optimization, many schemes are added to the conventional bi-directional evolutionary structural optimization (BESO) method in the previous studies. However, these schemes degrade the generality of BESO and increase the computational cost. This study proposes an improved topology optimization method for the continuum structures considering stress minimization in the framework of the conventional BESO method. A global stress measure constructed by p-norm function is treated as the objective function. To stabilize the optimization process, both qp-relaxation and sensitivity weight scheme are introduced. Design variables are updated by the conventional BESO method. Several 2D and 3D examples are used to demonstrate the validity of the proposed method. The results show that the optimization process can be stabilized by qp-relaxation. The value of q and p are crucial to reasonable solutions. The proposed sensitivity weight scheme further stabilizes the optimization process and evenly distributes the stress field. The computational efficiency of the proposed method is higher than the previous methods because it keeps the generality of BESO and does not need additional schemes.

Multi-Material Topology Optimization of Structures Using an Ordered Ersatz Material Model

This paper proposes a new element-based multi-material topology optimization algorithm using a single variable for minimizing compliance subject to a mass constraint. A single variable based on the normalized elemental density is used to overcome the occurrence of meaningless design variables and save computational cost. Different from the traditional material penalization scheme, the algorithm is established on the ordered ersatz material model, which linearly interpolates Young's modulus for relaxed design variables. To achieve a multi-material design, the multiple floating projection constraints are adopted to gradually push elemental design variables to multiple discrete values. For the convergent element-based solution, the multiple level-set functions are constructed to tentatively extract the smooth interface between two adjacent materials. Some 2D and 3D numerical examples are presented to demonstrate the effectiveness of the proposed algorithm and the possible advantage of the multi-material designs over the traditional solid-void designs.

RI-IGABEM in inhomogeneous heat conduction problems

The isogeometric analysis boundary element method (IGABEM) has great potential in the simulation of heat conduction problems due to its exact geometric representation and good approximation properties. In this paper, the radial integration IGABEM (RI-IGABEM) is proposed to solve isotropic heat conduction problems in inhomogeneous media with heat source. Similar to traditional BEM, the domain integrals cannot be avoided since the foundational solution for the Laplace equation is used to derive integral equation. In order to preserve the advantage of IGABEM, i.e. only boundary is discretized, the radial integration method (RIM) is applied to transform the domain integral into an equivalent boundary integral. In addition, using a simple transformation method, the uniform potential method is successfully applied to solve the strongly singular integrals, and the Telles scheme and the element sub-division method are used to solve the weakly singular integrals in RI-IGABEM respectively. In order to validate the accuracy and convergence of the RI-IGABEM in the analysis of the single or multiple boundary heat conduction problems, several 2D and 3D numerical examples are used to discuss the influence of some factors, such as the number of applied points, the order of basis functions, and the position of internal applied points.

Topological Design of Microstructures ofMaterials Containing Multiple Phases of Distinct Poisson’s Ratios

A methodology for achieving the maximum bulk or shear modulus in an elastic composite composed of two isotropic phases with distinct Poisson’s ratios is proposed. A topology optimization algorithm is developed which is capable of nding microstructures with extreme properties very close to theoretical upper bounds. The effective mechanical properties of the designed composite are determined by a numerical homogenization technique. The sensitivities with respect to design variables are derived by simultaneously interpolating Young’s modulus and Poisson’s ratio using different parameters. The so-called solid isotropic material with penalization method is developed to establish the optimization formulation. Maximum bulk or shear modulus is considered as the objective function, and the volume fraction of constituent phases is taken as constraints. The method of moving asymptotes is applied to update the design variables. Several 3D numerical examples are presented to demonstrate the effectiveness of the proposed structural optimization method. The effects of key parameters such as Poisson’s ratios and volume fractions of constituent phase on the nal designs are investigated. A series of novel microstructures are obtained from the proposed approach. It is found that the optimized bulk and shear moduli of all the studied composites are very close to the Hashin-Shtrikman-Walpole bounds.

Stability Analysis of Conewise Affine Dynamical Systems Using Conewise Linear Lyapunov Functions

This letter proposes computational algorithms for analyzing conewise affine dynamical systems, where every neighborhood of the origin contains an affine mode. These algorithms are based on conewise linear Lyapunov functions. To make such algorithms useful, we present an algorithm to automatically search over partitions defining these conewise Linear functions. This algorithm is sound, although we present a counter-example to its completeness. We show that this approach verifies stability of 2D and 3D examples of conewise affine dynamical systems, including combinations of the harmonic and nonsmooth oscillators.

Estimating posterior quantity of interest expectations in a multilevel scalable framework

AbstractScalable approaches for uncertainty quantification are necessary for characterizing prediction confidence in large‐scale subsurface flow simulations with uncertain permeability. To this end we explore a multilevel Monte Carlo approach for estimating posterior moments of a particular quantity of interest, where we employ an element‐agglomerated algebraic multigrid (AMG) technique to generate the hierarchy of coarse spaces with guaranteed approximation properties for both the generation of spatially correlated random fields and the forward simulation of Darcy's law to model subsurface flow. In both these components (sampling and forward solves), we exploit solvers that rely on state‐of‐the‐art scalable AMG. To showcase the applicability of this approach, numerical tests are performed on two 3D examples—a unit cube and an egg‐shaped domain with an irregular boundary—where the scalability of each simulation as well as the scalability of the overall algorithm are demonstrated.

Solving Monge-Ampère equation in 2D and 3D by Generalized Finite Difference Method

In this paper we derive the discretization of the nonlinear Monge-Ampère equation by means of the explicit formulae of the meshless Generalized Finite Difference Method (GFDM) in both two and three dimensional settings (2D and 3D). To do so we implement the Cascadic iterative algorithm. We provide a rigorous proof of the consistency of the GFDM for this elliptic equation and present several examples in 2D and 3D, where the method shows its potential and accuracy. We provide a discussion on the number of total nodes in the domain and the number of local supporting nodes. Also, we give an example with physical meaning where we find a surface whose Gaussian curvature is given.

Inverse design method of microscatterer array for realizing scattering field intensity shaping

It is a novel and interesting idea to inversely design the scattering structure with the desired scattering field intensity distribution in a given target area as the known information. The inverse design method proposed in this paper does not need to be optimized, and the spatial distribution and dielectric constant distribution of the micro-scatterer array can be quickly analytically calculated according to the desired scattering field intensity in the target area. First, based on the spatial Fourier transform and angular spectrum transformation, the plane wave sources required in all directions are inversely obtained from the electric field intensity distribution required in the target area. Then, based on the theory of induced source, a method of irradiating the array of all-dielectric micro-scatterers with incident electromagnetic field to generate the required plane wave source is proposed. The scattering fields generated by these micro-scatterers will be superimposed on the target area to achieve the desired scattering field strength intensity. Finally, according to the proposed inverse design theory model, a specific three-dimensional (3D) design is carried out. In the 3D example, we study the scattering field intensity distribution of the point-focused shape of the square surface target area, and show an all-dielectric micro-sphere distribution design. Its spatial distribution and permittivity distribution are both obtained through the rapid analytical calculation of the desired scattered field intensity shape in the target area. Finally, based on the principle of linear superposition, we quickly and easily generate the complex shapes of “I”, “T”, and “X” in the target area. The satisfactory results of full-wave simulation show that the proposed inverse design method is effective and feasible.

An efficient multi-resolution topology optimization scheme for stiffness maximization and stress minimization

This article develops a multi-resolution topology optimization (MTOP) approach based on the solid isotropic material with penalization (SIMP) method, which is effective in obtaining high-resolution designs at low computational cost. The extended finite element method (XFEM) is employed to decouple the analysis mesh, material description and nodal design variables. By the advantage of XFEM at modelling material discontinuity, detailed geometrical features are generated on a coarse analysis mesh. To obtain a clear interface between material grids, a variation of the traditional sensitivity filter is introduced to produce discrete solutions. The low computational costs make the proposed approach appropriate for dealing with problems requiring a high number of finite element analysis (FEA) processes, typically high-resolution/large-scale models, stress minimization, etc. Accurate von Mises stress is calculated on a high number of Gaussian points, making the approach perform better at stress minimization. Then, several 2D and 3D examples optimized by different solvers are illustrated to demonstrate the effectiveness and excellent generality of the proposed approach.

RI-IGABEM based on PIM in transient heat conduction problems of FGMs

The isogeometric analysis boundary element method (IGABEM) has a broad application prospect due to its exact geometric representation and good approximation properties. In this paper, a novel radial integration IGABEM (RI-IGABEM) based on precise integration method (PIM) is proposed to solve transient heat conduction problems of functionally gradient materials (FGMs) with heat source Similar to traditional boundary element method (BEM), in which the fundamental solution for the Laplace equation is used to derive the boundary-domain integral equations, the domain integrals caused by the heat source and the thermal conductivity varying with coordinates and the initial temperature appear in the transient heat conduction problems. In order to preserve the advantage of IGABEM, i.e. only boundary is discretized, the radial integration method (RIM) is applied to transform the domain integral into an equivalent boundary integral. In addition, using a simple transformation method, the uniform potential method is successfully applied to solve the strongly singular integrals, and the Telles scheme and the element sub-division method are used to solve the weakly singular integrals in RI-IGABEM respectively. Furthermore, in order to improve the stability of numerical results, the PIM is adopted to solve the time domain problem. In order to validate the accuracy and convergence of the RI-IGABEM 2D and 3D numerical examples are used to discuss the influence of some factors, such as the number and the position of applied points, the order of basis functions, the number of model refinements and the length of time step.

Robust concurrent topology optimization of multiscale structure under load position uncertainty

Concurrent topology optimization of macrostructure and microstructure has attracted significant interest due to its high structural performance. However, most of the existing works are carried out under deterministic conditions, the obtained design may be vulnerable or even cause catastrophic failure when the load position exists uncertainty. Therefore, it is necessary to take load position uncertainty into consideration in structural design. This paper presents a computational method for robust concurrent topology optimization with consideration of load position uncertainty. The weighted sum of the mean and standard deviation of the structural compliance is defined as the objective function with constraints are imposed to both macro- and micro-scale structure volume fractions. The Bivariate Dimension Reduction method and Gauss-type quadrature (BDRGQ) are used to quantify and propagate load uncertainty to calculate the objective function. The effective properties of microstructure are evaluated by the numerical homogenization method. To release the computation burden, the decoupled sensitivity analysis method is proposed for microscale design variables. The bi-directional evolutionary structural optimization (BESO) method is used to obtain the black-and-white designs. Several 2D and 3D examples are presented to validate the effectiveness of the proposed robust concurrent topology optimization method.

VDAC

We introduce carvable volume decomposition for efficient 3-axis CNC machining of 3D freeform objects, where our goal is to develop a fully automatic method to jointly optimize setup and path planning. We formulate our joint optimization as a volume decomposition problem which prioritizes minimizing the number of setup directions while striving for a minimum number of continuously carvable volumes, where a 3D volume is continuously carvable, or simply carvable, if it can be carved with the machine cutter traversing a single continuous path. Geometrically, carvability combines visibility and monotonicity and presents a new shape property which had not been studied before. Given a target 3D shape and the initial material block, our algorithm first finds the minimum number of carving directions by solving a set cover problem. Specifically, we analyze cutter accessibility and select the carving directions based on an assessment of how likely they would lead to a small carvable volume decomposition. Next, to obtain a minimum decomposition based on the selected carving directions efficiently, we narrow down the solution search by focusing on a special kind of points in the residual volume, single access or SA points, which are points that can be accessed from one and only one of the selected carving directions. Candidate carvable volumes are grown starting from the SA points. Finally, we devise an energy term to evaluate the carvable volumes and their combinations, leading to the final decomposition. We demonstrate the performance of our decomposition algorithm on a variety of 2D and 3D examples and evaluate it against the ground truth, where possible, and solutions provided by human experts. Physically machined models are produced where each carvable volume is continuously carved following a connected Fermat spiral toolpath.

Particle‐based Liquid Control using Animation Templates

AbstractIt is notoriously difficult for artists to control liquids while generating plausible animations. We introduce a new liquid control tool that allows users to load, transform, and apply precomputed liquid simulation templates in a scene in order to control a particle‐based simulation. Each template instance generates control forces that drive the global simulated liquid to locally reproduce the templated liquid behavior. Our system is augmented with a variable proportion of temporary particles to help efficiently reproduce the templated liquid density, with fewer requirements on the surrounding environment. The resulting control strategy adds only a small computational overhead, leading to quick visual feedback for resolutions allowing interactive simulation. We demonstrate the robustness and ease of use of our method on various examples in 2D and 3D.

Challenges in Moving to Multiscale Battery Models - Where Electrochemistrymeets and demands more from Math

As lithium-metal and lithium-sulfur batteries are receiving great attention being among the most promising candidates for next-generation batteries in transportation applications, there has been significant recent interest in studying their multiscale characteristics. Multiple studies have conveyed the importance of microscale effects, pressure effects, and morphology changes for these batteries, especially for batteries with lithium metal as the anode. Advances in computing power make researchers believe that the detailed multiscale models can be efficiently simulated to arrive at the insights for the degradation and performance loss; however, this is not true and special attention needs to be paid to local singularities, boundary layers, moving boundaries, etc. This work presents 2D examples that illustrate the importance of grid convergence studies, provides well-defined detailed models to test the efficiency of numerical schemes, and discusses the associated simulation challenges. Perspective on important numerical aspects for robustness and efficiency, and recommendations on the possible best practices for studies involving multiscale modeling are also discussed.

Challenges in Moving to Multiscale Battery Models - Where Electrochemistry Meets and Demands More from Math

There has been significant recent interest in studying multiscale characteristics of current and next-generation batteries, including lithium-metal and lithium-sulfur batteries. Multiple studies have conveyed the importance of microscale effects, pressure effects, and morphology changes. Advances in computing power make researchers believe that the detailed multiscale models can be efficiently simulated to arrive at the insights for the degradation and performance loss; however, this is not true and special attention needs to be paid to local singularities, boundary layers, moving boundaries, etc. This work presents 2D examples that illustrate the importance of grid convergence studies, provides well-defined detailed models to test the efficiency of numerical schemes, and discusses the associated simulation challenges. Perspective on important numerical aspects for robustness and efficiency, and recommendations on the possible best practices for studies involving multiscale modeling are also provided.