General Numerical Scheme Research Articles

Machine learning optimization of compact finite volume methods on unstructured grids

Fourier analysis based optimization techniques have been developed for numerical schemes on structured grids and result in significant performance improvements. However, there lacks an effective optimization technique for numerical schemes on general unstructured grids, due to the mesh geometry complexity that makes Fourier transformation infeasible. To address this issue, an optimization framework based on machine learning is developed in this paper for numerical schemes on unstructured grids. The optimization of a compact high-order variational reconstruction on triangular grids is performed to illustrate the framework. An artificial neural network (ANN) is used to predict the optimal values of the derivative weights on cell interfaces that are the free parameters of the variational reconstruction, given the local geometry as input. The ANN is trained by minimizing the solution errors of a linear advection equation on a set of generated compact stencils, with a set of sampled sine waves as initial conditions. The developed optimization framework is applicable to general numerical schemes on unstructured grids as the training does not require explicit computation of dispersion or dissipation. Numerical results for inviscid flow problems show that the optimized variational finite volume scheme is remarkably more accurate and efficient than its non-optimized counterpart, demonstrating the effectiveness of the machine learning optimization.

High order computation of optimal transport, mean field planning, and potential mean field games

Mean-field games (MFGs) have shown strong modeling capabilities for large systems in various fields, driving growth in computational methods for mean-field game problems. However, high order methods have not been thoroughly investigated. In this work, we explore applying general high-order numerical schemes with finite element methods in the space-time domain for computing the optimal transport (OT), mean-field planning (MFP), and potential MFG problems. We conduct several experiments to validate the convergence rate of the high order method numerically. Those numerical experiments also demonstrate the efficiency and effectiveness of our approach.

Continuous Time Limit of the Stochastic Ensemble Kalman Inversion: Strong Convergence Analysis

The Ensemble Kalman inversion (EKI) method is a method for the estimation of unknown parameters in the context of (Bayesian) inverse problems. The method approximates the underlying measure by an ensemble of particles and iteratively applies the ensemble Kalman update to evolve (the approximation of the) prior into the posterior measure. For the convergence analysis of the EKI it is common practice to derive a continuous version, replacing the iteration with a stochastic differential equation. In this paper we validate this approach by showing that the stochastic EKI iteration converges to paths of the continuous-time stochastic differential equation by considering both the nonlinear and linear setting, and we prove convergence in probability for the former, and convergence in moments for the latter. The methods employed can also be applied to the analysis of more general numerical schemes for stochastic differential equations in general.

The scaled boundary finite element method for dispersive wave propagation in higher‐order continua

AbstractThe classical theory of elasticity relies on the perfect structure of matter. No matter how small a medium is, it always stays homogeneous—but the reality is different. Even though the classical continuum mechanics suffices well for describing phenomena that evolve at super‐microscopic scales, it ceases to reproduce reasonable responses when the size effects are pronounced. It is due to the local constitutive equations of classical continuum mechanics, which are devoid of any length scales associated with the underlying microstructure. The gradient‐dependent theory of elasticity redresses this shortcoming by incorporating the kinematic quantities of the corresponding representative volume element by enriching the differential equations with higher‐order spatial and temporal derivatives. In this study, the scaled boundary finite element method is formulated for the equations of motion with higher‐order inertia terms. To this end, the semi‐discretized scaled boundary finite element equations of the medium are derived by introducing the scaled boundary transformation of geometry to the gradient‐enriched equations of motion and applying the Galerkin method of weighted residuals. It is shown that the well‐established available solution methods are incapable of handling the frequency‐domain representation of the derived formulation. Accordingly, a numerical solution method based on the shooting technique is proposed. The solution procedure is formulated for general numerical integration schemes via the infinite Taylor series. The evolution of the impedance‐diffusion matrix and contribution of inter‐subdomain forces and tractions are extracted. Four different numerical integration methods are employed to describe the solution procedure. In addition, a comparison regarding their computational efficiency is presented. For the sake of verification, the scaled boundary finite element solutions of three numerical examples are compared with reference solutions that are obtained using finite element models with extremely fine meshes. The convergence trends are also presented for both ‐ and ‐refinement methods. The results demonstrate the capability of the proposed formulation in reproducing accurate responses.

A general numerical scheme for the optimal control of fractional Birkhoffian systems

This paper gives a general numerical scheme for the optimal control problem of fractional Birkhoffian systems. The fractional forced Birkhoff equations within Riemann–Liouville fractional derivatives are derived from the fractional Pfaff–Birkhoff–d’Alembert principle which includes the control as an external force term. Following the strategy of variational integrators, the fractional Pfaff–Birkhoff–d’Alembert principle is directly discretized to develop the equivalent discrete fractional forced Birkhoff equations that served as the equality constraints of the optimization problem. Together with the initial and final state constraints on the configuration space, the original optimal control problem is converted into a nonlinear optimization problem subjected to a system of algebraic constraints, which can be solved by existing algorithms. An illustrative example is given to show the efficiency and simplicity of the proposed method.

Modeling escape from extinction with decomposable multi-type Sevastyanov branching processes

Biological populations under stress often face certain extinction unless they adapt toward unfavorable circumstances. In some scenarios such adaptation can assume the form of mutations within the genome of the population (e.g., cancer cells resisting chemotherapy, viruses developing resistance toward a vaccine), while in other scenarios adaptation can be in the form of movement toward some physical location (e.g., spreading of cancer cells to parts of the organism unaffected by treatment, animal populations fleeing polluted areas or areas struck by disaster). Regardless of the particular situation, it is often the case that cells/individuals with different levels of adaptation (which we may group into types) emerge among the cells/individuals of a stressed population. We propose a decomposable multi-type Sevastyanov branching process (possibly with multiple supercritical types) for modeling relevant aspects of the dynamics of such populations. The branching process developed within this paper is a generalization of the decomposable multi-type age-dependent branching process with a single supercritical type considered in Slavtchova-Bojkova and Vitanov. With respect to Slavtchova-Bojkova and Vitanov, we introduce additional, possibly supercritical, types into the interaction scheme between types, further, we incorporate possible dependence of the reproductive capabilities of cells/individuals from their age. We obtain a system of integral equations for the probability generating function of the new process and accordingly expand previous results from Slavtchova-Bojkova and Vitanov concerning probabilities of extinction, number of occurred mutations, waiting time to escape mutant, and immediate risk of escaping extinction. We also provide a general numerical scheme for calculating obtained systems of integral equations.

Machine Learning Enhanced Boundary Element Method: Prediction of Gaussian Quadrature Points

This paper applies a machine learning technique to find a general and efficient numerical integration scheme for boundary element methods. A model based on the neural network multi-classification algorithm is constructed to find the minimum number of Gaussian quadrature points satisfying the given accuracy. The constructed model is trained by using a large amount of data calculated in the traditional boundary element method and the optimal network architecture is selected. The two-dimensional potential problem of a circular structure is tested and analyzed based on the determined model, and the accuracy of the model is about 90%. Finally, by incorporating the predicted Gaussian quadrature points into the boundary element analysis, we find that the numerical solution and the analytical solution are in good agreement, which verifies the robustness of the proposed method.

Two-scale topology optimisation of cellular materials under mixed boundary conditions

This work proposes a theoretical/numerical framework for the topology optimisation of anisotropic architected cellular materials at different scales. In particular, the topological variable (i.e., the pseudo-density field) is defined at both the scale of the representative volume element (i.e., the unit cell) of the material and at the macroscopic scale of the structure. The two-scale topology optimisation problem is formulated in the most general sense, i.e., by considering non-zero Neumann-Dirichlet boundary conditions. The proposed method is based on: (a) non-uniform rational basis spline hyper-surfaces to represent the topological variable at each scale, (b) the solid isotropic material with penalisation approach, (c) a general numerical homogenisation scheme based on the strain energy to establish the link between scales. The proposed formulation exploits the properties of non-uniform rational basis spline entities to determine the relationships occurring among the topological variables defined at different scales to correctly state the optimisation problem and to satisfy the hypotheses at the basis of the homogenisation method. In particular, scale separation (a necessary condition to be met in order to apply the homogenisation method) and manufacturing requirements are implicitly ensured by introducing minimum length scale constraints on the topological variables defined at both macroscopic scale and unit cell scale, respectively. Furthermore, the sensitivity of the optimised topology (at each scale) to the applied boundary conditions and to the elastic symmetry group of the representative volume element is investigated by founding new and original results. The effectiveness of the approach is tested on 2D and 3D benchmark problems taken from the literature.

Thermal design of graded architected cellular materials through a CAD-compatible topology optimisation method

Architected cellular materials (ACMs) with a periodic micro-structure are often employed in high-performance components obtained through additive manufacturing (AM) technologies due to their high specific strength and stiffness. ACMs are also used in thermal applications, where their high surface-to-mass ratio can be conveniently exploited to enhance heat transfer. In this work, a numerical approach to predict the effective thermal conductivity (ETC) of ACMs obtained by AM is proposed. The model is based on a general numerical homogenisation scheme and an explicit description of the representative volume element (RVE) of the ACM. Numerical analyses have been conducted on 31 RVEs geometries: results show that the macroscopic ETC of ACMs strongly depends on the relative density and the geometrical features of the RVE. Moreover, starting from the database of RVEs geometries, seven configurations are chosen to design graded ACMs through a computer-aided design-compatible topology optimisation method based on non-uniform rational basis spline hyper-surfaces to represent the pseudo-density field, and on the well-known solid isotropic material with penalisation (SIMP) approach. In particular, the penalisation law used in the SIMP method is replaced by a physically-based penalisation scheme obtained by interpolating the results of the homogenisation for each RVE topology and a suitable post-processing phase is developed to recover the distribution of the graded ACM over the structure from the results of the optimisation process. The effectiveness of the proposed approach is shown on 2D and 3D benchmark problems taken from the literature.

Effect of Viscous Damping Models on Displacement Ductility Demands for SDOF Systems

The viscous damping is a highly idealized but mathematically convenient way of representing the energy dissipation mechanisms not related to the hysteretic response. In this study, the tangent-stiffness proportional Rayleigh damping (TSPRD) appropriate for inelastic hysteretic structures outfitted with the supplemental damping devices is utilized in inelastic dynamic analysis of simple single-degree-of-freedom (SDOF) structures. A numerical procedure of dynamic equilibrium equation for SDOF systems with TSPRD model is derived for development of the general numerical solution scheme. By specifying various combinations of tangent-stiffness and mass proportional damping terms in TSPRD model, the influence of viscous damping models on displacement ductility demands is investigated. The results indicate that the difference in ductility demands for various damping models is considerable, depending on the periods of vibration, relative lateral strength, hysteretic models, stiffness deterioration and initial damping ratios. The constant-strength displacement ductility spectra, by considering both tangent-stiffness proportional and mass proportional damping, are developed in terms of site conditions, hysteretic rules and initial damping ratios.

Screw dislocations around voids of any shape: A generalized numerical approach

Analytical solutions for dislocations around multiple interacting free surfaces are strictly case-specific (they depend on the exact dislocation location and number and geometry of voids/free surface). This is because they rely on the precise placement of image (fictitious) dislocations, and thus can end up being very complicated in their formulation. The current article presents a general numerical scheme based on the collocation point method. The proposed method is capable of generating solutions for dislocations around multiple interacting free surfaces of any shapes or geometries. The paper also furnishes verification of the developed methodology and new results.

Stability of numerical solutions for the stochastic pantograph differential equations with variable step size

In this paper, two different techniques are used to research the stability of numerical solutions with variable step size for stochastic pantograph differential equations. Based on the discrete semimartingale convergence theorem, the almost sure polynomial stability and the αth moment polynomial stability of the numerical solutions by the Euler–Maruyama (EM) and the backward Euler–Maruyama (BEM) methods with variable step size have been discussed. By using the discrete Razumikhin-type technique, the global αth moment asymptotically stability and the αth moment polynomial stability of the numerical solutions by the general numerical scheme with variable step size have been researched. Based on the results of the stability of the general numerical scheme, we discuss the stability of two special numerical solutions, namely the EM method and the BEM method. By comparing the two techniques (i.e. the discrete semimartingale convergence theorem and the discrete Razumikhin-type technique) on the research of the moment stability, we can easily get the latter is better. To illustrate the theoretical results, we give some examples to examine the almost sure polynomial stability and the αth moment polynomial stability of the numerical solutions by the EM and BEM schemes with variable step size.

An experimental and numerical study of the high cycle multiaxial fatigue strength of titanium lattice structures produced by Selective Laser Melting (SLM)

A numerical approach is proposed to assess the high cycle fatigue (HCF) strength of periodic cellular structures produced by Selective Laser Melting under multiaxial loads. The model is based on a general numerical homogenisation scheme and an explicit description of the Elementary Cell (EC) combined to an extreme values analysis making use of a fatigue indicator parameter based on Crossland’s criterion. This method is applied to 33 EC topologies.In addition, geometric discrepancy and surface roughness are experimentally characterised and considered in the numerical model using three methods which are compared to the experimental HCF strength.

A class of approximate computation method for antenna directivity

Directivity is one of the core performance parameters for an antenna, and its accurate computation and measurement have been receiving attention in the past decades. While quite a number of computational methods for antenna directivity are developed, a comprehensive comparison among these methods, including their advantages and disadvantages, has not yet been reported. In the literature, most of these methods can be roughly classified as two categories, i.e. numerical methods and analytical methods. The numerical methods include those simplified numerical methods for accessing the radiation pattern. For example, some methods assume that the radiation pattern is only elevation-angle dependent, and an approximate truncation of the integral interval is often employed. The analytical methods are developed, where the electromagnetic radiation intensity and the total radiation power for a specific antenna are derived analytically. Nevertheless, so far there have been less efforts dedicated to developing general computational methods based on the antenna's planar near-field theory and associated computational schemes, although indeed quite a few of advanced mathematical approaches have been used to obtain the accurate directivity, including the series expansion based on the spherical wave scheme and the Fourier expansion scheme. In spite of these efforts, one can see that a general accurate numerical scheme for the antenna directivity is still urgently required. As an important supplement to these efforts, this paper adopts or proposes four numerical integration methods regarding planar near-field measurement of the antennas under test. Based on the planar near-field theory, microwave near-field data, and fast Fourier transform (FFT), this paper discusses the numerical integration-based algorithm schemes based on the simulated (or measured) near-field data and relevant numerical uncertainty estimation. Specifically, an antenna with accurate directivity is chosen for illustrating the accuracy assessment. First, a set of near-field data under the well-established probing condition is simulated and then used as an input for subsequent numerical calculation of the antenna directivity value. Four different numerical integration methods are employed and their possible numerical errors are discussed. Consequently, the antenna directivity values are obtained respectively from the four different numerical methods. It is revealed that the numerically evaluated directivity values from the near-field data are roughly at most 0.6 dB larger than the accurate value. This work represents a substantial step toward a reliable estimation of the antenna directivity from the near-field data of an antenna.

Determination of the effective thermoelastic properties of cork-based agglomerates

In this paper, a general numerical homogenisation scheme coupled with an efficient modelling strategy for predicting the effective thermoelastic properties of cork-based agglomerates is presented. ...

A stabilized well-balanced RBF-FD meshless method for shallow-water equations

In this work, we are concerned with the study and computing of stabilized radial basis function-generated finite difference (RBF-FD) approximations for shallow-water equations. In order to obtain both stable and highly accurate numerical approximations of convection-dominated shallow-water equations, we use stabilized flat Gaussians (RBFSGA-FD) and polyharmonic splines with supplementary polynomials (RBFPHS-FD) as basis functions, combined with modified method of characteristics. These techniques are combined with careful design for the spatial derivative operators in the momentum flux equation, according to a general criterion for the exact preservation of the “lake at rest” solution in general mesh-based and meshless numerical schemes for the strong form of the shallow-water equations with bottom topography. Both structured and unsructured point clouds are employed for evaluating the influence of cloud refinement, size of local supports and maximal permissible degree of the polynomials in RBFPHS-FD.

Determination of the effective elastic properties of titanium lattice structures

This work aims at proposing a numerical approach to predict the mechanical response of lattice structures fabricated by means of a suitable additive layer manufacturing (ALM) process. The numerical procedure makes use of an explicit description of the lattice representative volume element (RVE) to investigate and compare the influence of the lattice RVE topology on the macroscopic elastic response. The proposed method is based on a general numerical homogenization scheme to determine the effective elastic properties of the lattice RVE at the macroscopic scale. A large numerical campaign has been conducted on 17 geometries. The effect of the RVE local features, e.g., the presence of a fillet radius and/or of a spherical node between struts composing the RVE as well as the surface roughness, on the effective elastic properties is investigated. Results show that both the relative density and the geometrical features of the RVE strongly affect the equivalent macroscopic elastic behavior of the lattice.

Numerical Solution of Dispersive Optical Solitons with Schrödinger‐Hirota Equation by Improved Adomian Decomposition Method

In this paper, we present new numerical results for the dispersive optical soliton solutions of the nonlinear Schrödinger‐Hirota equation. The spatio‐temporal dispersion term is included, in addition to group velocity dispersion Kerr law of nonlinearity are studied. A general recursive numerical scheme for the equation is devised via the Improved Adomian Decomposition Method (IADM) and further sought for some analytical results for validation. The scheme is shown to be efficient and possessed high level of accuracy as demonstrated.

Multi-axial fatigue behaviour of titanium periodic cellular structures produced by Selective Laser Melting (SLM)

Additive manufacturing techniques such as selective laser melting (SLM) allow for manufacturing periodic porous titanium structures having complex geometry. The mechanical properties of these structures with different elementary cell (ECs) designs have been already studied in literature. However, their fatigue behaviour is not yet well understood. This work aims at proposing a numerical approach to predict the periodic cellular structures fatigue behaviour under multiaxial loadings. The approach is based on an explicit description of the EC combined to an extreme statistical analysis making use of a fatigue indicator parameter to investigate (and compare) different EC configurations. On the other hand, the numerical model relies on the use of a general numerical homogenisation scheme to properly apply the multi-axial loading conditions to the EC at the mesoscopic scale. Results show that lattice fatigue strength is strongly affected by the relative density as well as by the geometrical features of the EC.

Multidimensional mixed–hybrid finite element method for compositional two-phase flow in heterogeneous porous media and its parallel implementation on GPU

A general multidimensional numerical scheme, primarily designed to simulate two-phase compositional flow in porous media, is presented with serial and parallel implementations suitable for solving problems with degenerate (capillary) diffusion or capillary barrier effect in heterogeneous porous materials. The numerical scheme is based on the mixed–hybrid finite element method with the semi-implicitapproach for the time discretization in order to obtain a system of linear equations in each time step. The scheme is implemented in serial for CPU and in parallel for CPU and GPU using TNL that provides an efficient abstraction layer for accessing various parallel hardware architectures. In order to demonstrate the applicability of the numerical scheme, a numerical analysis is presented for problems of two-phase flow in 1D, 2D, and 3D for which exact (semi-analytical) solutions are known and a series of benchmark problems for two-phase flow in heterogeneous porous media is discussed to show correct simulation of the capillary barrier effect. The efficiency and accuracy of the implementations on CPU and GPU are discussed. Moreover, we construct an analytical solution and use it to demonstrate convergence of the numerical scheme for two-phase compositional flow problems in porous media.