Classical Finite Element Research Articles

Influence of foundation flexibility on the topology optimization of piled structures

AbstractThis article investigates the influence of foundation and soil flexibility on the topology optimization of piled structures. The structure is modeled with classical finite elements, while the embedded pile group foundation and its interaction with the soil are modeled via the stiffness matrix method. Coupling between the structure and its foundation is obtained by establishing direct continuity and equilibrium conditions at their interface. Topology optimization of the discretized piled structure is performed through the Bi‐directional Evolutionary Structural Optimization (BESO) method for various boundary conditions. The objective function of the optimization problem is the minimization of the compliance of the piled structure under prescribed volume restriction. The results show that the flexibility of the soil and piles may have a significant effect on the optimal topology of the structure, as well as on the achievable optimization objective.

A Virtual Element Method for the Wave Equation on Curved Edges in Two Dimensions

In this work we present an extension of the Virtual Element Method with curved edges for the numerical approximation of the second order wave equation in a bidimensional setting. Curved elements are used to describe the domain boundary, as well as internal interfaces corresponding to the change of some mechanical parameters. As opposite to the classic and isoparametric Finite Element approaches, where the geometry of the domain is approximated respectively by piecewise straight lines and by higher order polynomial maps, in the proposed method the geometry is exactly represented, thus ensuring a highly accurate numerical solution. Indeed, if in the former approach the geometrical error might deteriorate the quality of the numerical solution, in the latter approach the curved interfaces/boundaries are approximated exactly guaranteeing the expected order of convergence for the numerical scheme. Theoretical results and numerical findings confirm the validity of the proposed approach.

Partition of Unity Finite Element Method for 2D Vibro-Acoustic Modeling

The Partition of Unity Finite Element Method (PUFEM) shows promise for modeling wave-like problems in the mid-to-high frequency range, allowing to capture several wavelengths in a single element. Despite the increasing attention it received in acoustics and in structural dynamics, its efficacy to deal with coupled problems has not been addressed. The main challenge in this case is to be able to represent different types of physical waves accurately, knowing that the wavelengths can be very different and vary differently, exemplified by the dispersion of flexural waves in a solid. Without a proper handling of the coupling between the coupled media, at best the number of degrees of freedom (DoF) will not be optimal, at worst the coupled model will not converge. Techniques like mesh refinement, wave enrichment and compatible or incompatible meshes might offer a potential solution to the problem, but the model usually needs to be adjusted through a time consuming trial-and-error procedure. To tackle the problem, this paper considers a 2D coupled vibro-acoustic problem, in which the structural and acoustic domains, modeled with PUFEM, are coupled using compatible and incompatible meshes based on different coupling strategies. Numerical analyses show that the proposed method outperforms the classical finite element method by several orders of magnitude in terms of number of DoF. Recommendations are proposed on the technique to choose depending on the frequency range of interest in relation to the critical frequency of the structure to ensure the best convergence rate. Finally, an application example is presented to highlight the performance of the proposed method.

A model order reduction technique for FFT‐based microstructure simulation using a geometrically adapted reduced set of frequencies

AbstractThe FFT‐based method introduced by Moulinec and Suquet [9] serves as an alternative for the classical finite element based simulation of periodic microstructures. This simulation approach makes use of fast Fourier transforms (FFT) as well as fixed‐point iterations to solve the microscopic boundary value problem which is captured by the Lippmann‐Schwinger equation. Kochmann et al. [5] introduced a model order reduction technique using a reduced set of frequencies to decrease the computational effort of solving the Lippmann‐Schwinger equation in Fourier space. This earlier proposed method is based on a fixed sampling pattern, which determines the reduced set of frequencies. Instead of the fixed sampling pattern, we propose to use a geometrically adapted choice of frequencies, which corresponds to the representation of phases within the considered microstructure.

A Mixed Acoustic Fluid Finite Element in Boundary Representation for Fluid‐Structure Interaction

AbstractThis contribution focuses on a mixed finite element formulation for an acoustic fluid in boundary representation to model fluid‐structure interaction problems. In fact, acoustic fluids are characterized by their lack of dynamic viscosity and undergo displacements through a pure pressure wave propagation. Since many fluids, for instance water, show nearly‐incompressible behavior, volumetric locking occurs in the classical finite element formulation. This drawback is herein evaded by a two‐field, or mixed, finite element formulation. To facilitate the contact condition at the fluid‐structure interface, the acoustic fluid is expressed by structural field variables, which are nodal displacements and internal pressures. Both the displacements and internal pressures are parameterized by a Scaled‐Boundary (SB) approach, in which an interpolation in the scaling direction is employed. From this parametrization, the element mass and stiffness matrices of the fluid can be formulated. Due to the special parametrization of the element boundary, pressure continuity is required between scaled sections in the element interior, although pressure continuity may be interrupted between neighboring Scaled‐Boundary elements. Since the lack of shear viscosity leads to unwanted zero‐energy modes concerned with the fluid's vorticity, this vorticity is penalized on a sectional level. As a result, these modes do not show up in the frequency spectrum of the dynamic eigenvalue problem any longer. The natural frequencies and mode shapes in this spectrum deliver accurate results, even for very coarse meshes.

A chemorepulsion model with superlinear production: analysis of the continuous problem and two approximately positive and energy-stable schemes

We consider the following repulsive-productive chemotaxis model: find u ≥ 0, the cell density, and v ≥ 0, the chemical concentration, satisfying1∂tu−Δu−∇⋅(u∇v)=0inΩ,t>0,∂tv−Δv+v=upinΩ,t>0,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$ \\left\\{ \\begin{array}{l} \\partial_t u - {\\Delta} u - \\nabla\\cdot (u\\nabla v)=0 \\ \\ \\text{ in}\\ {\\Omega},\\ t>0,\\\\ \\partial_t v - {\\Delta} v + v = u^p \\ \\ { in}\\ {\\Omega},\\ t>0, \\end{array} \\right. $$\\end{document}with p ∈ (1, 2), {Omega }subseteq mathbb {R}^{d} a bounded domain (d = 1, 2, 3), endowed with non-flux boundary conditions. By using a regularization technique, we prove the existence of global in time weak solutions of (1) which is regular and unique for d = 1, 2. Moreover, we propose two fully discrete Finite Element (FE) nonlinear schemes, the first one defined in the variables (u,v) under structured meshes, and the second one by using the auxiliary variable σ = ∇v and defined in general meshes. We prove some unconditional properties for both schemes, such as mass-conservation, solvability, energy-stability and approximated positivity. Finally, we compare the behavior of these schemes with respect to the classical FE backward Euler scheme throughout several numerical simulations and give some conclusions.

Uniform convergent modified weak Galerkin method for convection-dominated two-point boundary value problems

We propose and analyze a modified weak Galerkin finite element method (MWG-FEM) for solving singularly perturbed problems of convection-dominated type. The proposed method is constructed over piecewise polynomials of degree $k\geq1$ on interior of each element and piecewise constant on the boundary of each element. The present method is parameter-free and has less degrees of freedom compared to the classical weak Galerkin finite element method. The method is shown uniformly convergent for small perturbation parameters. An uniform convergence rate of $\mathcal {O}((N^{-1}\ln N)^k)$ in the energy-like norm is established on the piecewise uniform Shishkin mesh, where $N$ is the number of elements. Various numerical examples are presented to confirm the theoretical results. Moreover, we numerically confirm that the proposed method has the optimal order error estimates of $\mathcal {O}(N^{-(k+1)})$ in a discrete $L^2$-norm and converges at superconvergence order of $\mathcal {O}((N^{-1}\ln N)^{2k})$ in the discrete $L^\infty$-norm.

A finite element method with cover functions for underwater acoustic propagation problems

To improve the performance of the classical finite element method (FEM) in acoustic analyses, in this work the interpolation cover functions are incorporated into the standard linear nodal shape functions to give the finite element with cover functions (FEwC). Since the original linear approximation space of the standard FEM is significantly enriched, more accurate solutions can be obtained. However, this FEwC encounters the intractable linear dependence (LD) problem. In this paper the LD problem is studied systematically and then an effective method is given to completely avoid this problem. The imposition of Dirichlet boundary conditions in FEwC which is different from that in FEM is also illustrated. Besides, to handle the acoustic scattering problems in the unbounded domains, the Dirichlet-to-Neumann (DtN) map technique is employed as the non-reflecting boundary condition. The dispersion error analyses and several typical numerical experiments associated with the joint frequency-domain and time-domain analysis are performed to exhibit the high performance of the FEwC compared with the classic FEM and the edge-based smoothed FEM in solving underwater acoustic propagation problems.

The synthesized method based on classical mechanics and finite element for vehicle collision accident reconstruction analysis

ABSTRACT Most of the traditional methods for accident reconstruction are only based on a single theoretical system – either classical mechanics or finite element. In order to compensate for this limitations, the analytic methods of finite element and classical mechanics were synthesized to simulate and reconstruct a vehicular collision and deformation. Using an actual accident case, the finite element simulation models of accident vehicles are constructed, and the simulation experiment is also conducted. During the experiment, there are some analyses about the characteristics of velocity variation, forming process of deformation, state of stress distribution, and law of energy transformation of vehicles, and analyses also including vehicle simulation deformation and actual deformation shapes. It is shown that the synthesized method breaks through the limitations of traditional methods that rely on a single theoretical system. Meanwhile, comparing the finite element simulation deformation with the actual deformation can improve the reliability of accident reconstruction.

Modeling variable-density flow in saturated-unsaturated porous media: An advanced numerical model

Modeling variable-density flow in unconfined aquifers is a challenging task because of the nonlinear coupling between variably saturated flow and contaminant transport. This results in a highly nonlinear system since the strongly nonlinear Richards flow equation is, in addition, coupled to the advection-dispersion transport equation by viscosity and density variation. The solution of such a nonlinear system is often subject to convergence issues and can be very expansive in terms of computational time, especially for large-scale problems. Conventional numerical algorithms based on the sequential approach and the classical finite difference or finite element methods with the first-order backward Euler time integration scheme are generally inefficient and/or do not provide satisfactory results. In this work, we develop a new efficient and accurate 2D numerical model for the transport of dense contaminants in unsaturated porous media that allows for the simulation of large-scale problems. This research describes a new model that combines advanced spatial discretization methods (mixed hybrid finite element method, discontinuous Galerkin finite element method, and multipoint flux approximation method) with higher-order time integration techniques via the method of lines (MOL). The latter allows one to adapt the time step's size and the order of the time integration to improve the computational efficiency while maintaining accuracy. The robustness and accuracy of the new model are shown by comparison against a widely used commercial code based on the standard finite element method. The applicability of the developed model to a large-scale problem is then investigated by simulating saltwater intrusion under a climate change projection and long-term pumping regimes for the Akkar coastal aquifer in Lebanon using a simplified 2D conceptual model.

Dynamic stability optimization of laminated composite plates: An isogeometric HSDT formulation and PSO algorithm

An isogeometric analysis framework based on the non-uniform rational B-spline (NURBS) and the higher-order shear deformation theory (HSDT) is developed to study the geometry-accurate dynamic instability behavior of the shear-sensitive generally laminated composite plates. The developed approach is combined with the particle swarm optimization (PSO) technique to modify the layup orientations of the laminate corresponding to the minimum dynamic instability opening (DIOI) value. Based on the accuracy assessment results, the convergence rate of the developed method is higher than the classical finite element method. Moreover, the frequency accuracy of cubic NURBS is stabilized for high degrees of freedom denoting their high computational efficiency. The optimization procedure reveals that the mode shapes of the plates with optimal layups are more complex than their non-optimal case. The optimal ply angles are mostly oriented along the maximum-curvature path, perpendicular to the loading direction, and parallel to the supported edges. The mixed-type supports leads to lower DIOI values for optimal layups concerning the homogeneous supports. The pure compressive force expands the fundamental dynamic instability region (DIRI) for the generally laminated plates and leads to wide-range frequency instabilities. However, the combination of compressive force with lateral or shear force increases the lower bound of the DIRI for optimal layups. The minimization of DIOI value does not necessarily increase the lower bound of DIRI for optimal layups.

A modified rate-dependent peridynamic model with rotation effect for dynamic mechanical behavior of ceramic materials

As a mathematical expression of the dynamic mechanical behavior, the constitutive model plays an indispensable role in numerical simulations of ceramic materials. The current bond-based peridynamic constitutive models can accurately describe the dynamic mechanical behavior of partial ceramic materials under impact loading, however, the predicted value of the Poisson’s ratio is 0.25, which is not true for most of the known ceramic materials. Herein, based on the existing bond-based peridynamic constitutive model, the current study utilizes the description of tangential bond force and considers the influence of bond force on rotation to accurately predict the Poisson’s ratio of different types of ceramic materials, e.g., SiC and B4C. Moreover, a novel and comprehensive damage assessment criterion during tensile–shear​ and compressive–shear is established, and the rate-dependent bond-based peridynamic constitutive model is improved, which realized the optimization of the original model. Then, the improved rate-dependent bond-based peridynamic constitutive model is employed to conduct several numerical experiments, including frontal impact, edge-on impact and uniaxial compression experiments. The simulation results are compared with the classical finite element model and experimental results, demonstrating the superiority of the optimized model in terms of accuracy and versatility. The optimized model describes the damage behavior of different ceramic materials and provides theoretical bases for future work in penetration behavior of ceramics.

Radial lattice quantization of 3D ϕ4 field theory

The quantum extension of classical finite elements, referred to as quantum finite elements ({\bf QFE})~\cite{Brower:2018szu,Brower:2016vsl}, is applied to the radial quantization of 3d $\phi^4$ theory on a simplicial lattice for the $\mathbb R \times \mathbb S^2$ manifold. Explicit counter terms to cancel the one- and two-loop ultraviolet defects are implemented to reach the quantum continuum theory. Using the Brower-Tamayo~\cite{Brower:1989mt} cluster Monte Carlo algorithm, numerical results support the QFE ansatz that the critical conformal field theory (CFT) is reached in the continuum with the full isometries of $\mathbb R \times \mathbb S^2$ restored. The Ricci curvature term, while technically irrelevant in the quantum theory, is shown to dramatically improve the convergence opening, the way for high precision Monte Carlo simulation to determine the CFT data: operator dimensions, trilinear OPE couplings and the central charge.

An Arbitrary-Order Discrete de Rham Complex on Polyhedral Meshes: Exactness, Poincaré Inequalities, and Consistency

In this paper, we present a novel arbitrary-order discrete de Rham (DDR) complex on general polyhedral meshes based on the decomposition of polynomial spaces into ranges of vector calculus operators and complements linked to the spaces in the Koszul complex. The DDR complex is fully discrete, meaning that both the spaces and discrete calculus operators are replaced by discrete counterparts, and satisfies suitable exactness properties depending on the topology of the domain. In conjunction with bespoke discrete counterparts of text {L}^2-products, it can be used to design schemes for partial differential equations that benefit from the exactness of the sequence but, unlike classical (e.g., Raviart–Thomas–Nédélec) finite elements, are nonconforming. We prove a complete panel of results for the analysis of such schemes: exactness properties, uniform Poincaré inequalities, as well as primal and adjoint consistency. We also show how this DDR complex enables the design of a numerical scheme for a magnetostatics problem, and use the aforementioned results to prove stability and optimal error estimates for this scheme.

Discontinuous Galerkin methods through the lens of variational multiscale analysis

In this article, we present a theoretical framework for integrating discontinuous Galerkin methods in the variational multiscale paradigm. Our starting point is a projector-based multiscale decomposition of a generic variational formulation that uses broken Sobolev spaces and Lagrange multipliers to accommodate the non-conforming nature at the boundaries of discontinuous Galerkin elements. We show that existing discontinuous Galerkin formulations, including their penalty terms, follow immediately from a specific choice of multiscale projector. We proceed by defining the “fine-scale closure function”, which captures the closure relation between the remaining fine-scale term in the discontinuous Galerkin formulation and the coarse-scale solution via a single integral expression for each basis function in the coarse-scale test space. We show that the projectors that correspond to discontinuous Galerkin methods lead to fine-scale closure functions with more compact support and smaller amplitudes compared to the fine-scale closure function of the classical (conforming) finite element method. This observation provides a new perspective on the natural stability of discontinuous Galerkin methods for hyperbolic problems, and may open the door to rigorously designed variational multiscale based fine-scale models that are suitable for DG methods.

Discontinuous Galerkin for the wave equation: a simplified a priori error analysis

Standard discontinuous Galerkin methods, based on piecewise polynomials of degree , are considered for temporal semi-discretization for second order hyperbolic equations. The main goal of this paper is to present a simple and straightforward a priori error analysis of optimal order with minimal regularity requirement on the solution. Uniform norm in time error estimates are also proved. To this end, energy identities and stability estimates of the discrete problem are proved for a slightly more general problem. These are used to prove optimal order a priori error estimates with minimal regularity requirement on the solution. The combination with the classic continuous Galerkin finite element discretization in space variable is used, to formulate a full-discrete scheme. The a priori error analysis is presented. Numerical experiments are performed to verify the theoretical results.

A new finite element approach to model microscale strain localization within olivine aggregates

Abstract. This paper presents a new mesoscopic full field approach for the modeling of microstructural evolutions and mechanical behavior of olivine aggregates. The mechanical framework is based on a reduced crystal plasticity (CP) formulation which is adapted to account for non-dislocation glide strain-accommodating mechanisms in olivine polycrystals. This mechanical description is coupled with a mixed velocity–pressure finite element (FE) formulation through a classical crystal plasticity finite element method (CPFEM) approach. The microstructural evolutions, such as grain boundary migration and dynamic recrystallization, are also computed within a FE framework using an implicit description of the polycrystal through the level-set approach. This numerical framework is used to study the strain localization, at the polycrystal scale, on different types of pre-existing shear zones for thermomechanical conditions relevant to laboratory experiments. We show that both fine-grained and crystallographic textured pre-existing bands favor strain localization at the sample scale. The combination of both processes has a large effect on strain localization, which emphasizes the importance of these two microstructural characteristics (texture and grain size) on the mechanical behavior of the aggregate. Table 1 summarizes the list of the acronyms used in the following.

Generalized higher order layerwise theory for the dynamic study of anisotropic doubly -curved shells with a mapped geometry

An innovative generalized Layer-Wise (LW) approach is proposed for the modal analysis of anisotropic doubly-curved shells with an arbitrary shape by employing the Generalized Differential Quadrature (GDQ) method. The geometrical description of shell structures can be traced from both the differential geometry and mapping technique based on a proper definition of the blending functions.The kinematic field is described in each lamina by employing several interpolating polynomials in a generalized setting, while developing a LW theory that involves different orders of kinematic expansion. The Equivalent Single Layer (ESL) generalized formulation is, thus, derived as particular case of the LW theory. Angle-ply generally anisotropic laminated shells are here investigated, as well as layups with a thick foam softcore. Several numerical examples are illustrated, including different boundary conditions of the mapped geometry, different geometric curvatures for both unmapped and mapped paths, along with a variety of interpolating polynomials and expansion orders of the kinematic field. The accuracy of the proposed model is well-established by a comparative evaluation between our solutions in terms of frequencies and mode shapes and predictions based on classical 3D finite elements.

A numerical homogenization technique for unidirectional composites using polygonal generalized finite elements

The present study proposes a computational methodology to obtain the homogenized effective elastic properties of unidirectional fibrous composite materials by using the generalized finite-element method and penalization techniques to impose periodic boundary conditions on non-uniform polygonal unit cells. Each unit cell is described by a single polygonal finite element using Wachspress functions as base shape functions and different families of enrichment functions to account for the internal fiber influence on stresses and strains fields. The periodic boundary conditions are imposed using reflection laws between two parallel opposing faces using a Lagrange multiplier approach; this reflection law creates a distributed reaction force over the edges of the [Formula: see text]-gon from the direct application of a given deformation gradient, which simulates different macroscopic load cases on the macroscopic body the unit cell is part of. The methodology is validated through a comparison with results for similar unit cells found in the literature and its computational efficiency is compared to simple cases solved using a classic finite-element approach. This methodology showed computational advantages over the classic finite elements in both computational efficiency and total number of degrees of freedom for convergence and flexibility on the shape of the unit cell used. Finally, the methodology provides an efficient way to introduce non-circular fiber shapes and voids.

A Massively Parallel Hybrid Finite Volume/Finite Element Scheme for Computational Fluid Dynamics

In this paper, we propose a novel family of semi-implicit hybrid finite volume/finite element schemes for computational fluid dynamics (CFD), in particular for the approximate solution of the incompressible and compressible Navier-Stokes equations, as well as for the shallow water equations on staggered unstructured meshes in two and three space dimensions. The key features of the method are the use of an edge-based/face-based staggered dual mesh for the discretization of the nonlinear convective terms at the aid of explicit high resolution Godunov-type finite volume schemes, while pressure terms are discretized implicitly using classical continuous Lagrange finite elements on the primal simplex mesh. The resulting pressure system is symmetric positive definite and can thus be very efficiently solved at the aid of classical Krylov subspace methods, such as a matrix-free conjugate gradient method. For the compressible Navier-Stokes equations, the schemes are by construction asymptotic preserving in the low Mach number limit of the equations, hence a consistent hybrid FV/FE method for the incompressible equations is retrieved. All parts of the algorithm can be efficiently parallelized, i.e., the explicit finite volume step as well as the matrix-vector product in the implicit pressure solver. Concerning parallel implementation, we employ the Message-Passing Interface (MPI) standard in combination with spatial domain decomposition based on the free software package METIS. To show the versatility of the proposed schemes, we present a wide range of applications, starting from environmental and geophysical flows, such as dambreak problems and natural convection, over direct numerical simulations of turbulent incompressible flows to high Mach number compressible flows with shock waves. An excellent agreement with exact analytical, numerical or experimental reference solutions is achieved in all cases. Most of the simulations are run with millions of degrees of freedom on thousands of CPU cores. We show strong scaling results for the hybrid FV/FE scheme applied to the 3D incompressible Navier-Stokes equations, using millions of degrees of freedom and up to 4096 CPU cores. The largest simulation shown in this paper is the well-known 3D Taylor-Green vortex benchmark run on 671 million tetrahedral elements on 32,768 CPU cores, showing clearly the suitability of the presented algorithm for the solution of large CFD problems on modern massively parallel distributed memory supercomputers.