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The scaled boundary finite element method is known for its capability in reproducing highly-detailed solution fields over large subdomains, and also for its flexibility in terms of spatial discretization, enabling us to define subdomains with arbitrary numbers of boundary elements and hanging nodes. The former, however, is not readily attainable in the cases whose closed-form solution does not exist. Those cases invoke the use of a cascade of expensive computations for the recovery of inner-subdomain solution fields, leaving one of the main assets of the method almost unused. As a remedy, we propose a new solution scheme by which the interior fields of subdomains can be easily recovered so that the full potential of the scaled boundary finite element method can be harvested. No auxiliary variables are introduced to the global algebraic system and the dimensions of the matrices remain constant. Most of the computations are carried out prior to the global algebraic system assembly, and in a completely decoupled manner on the element level, rendering the introduced method to be embarrassingly parallelizable. The inner-subdomain information is not required for the solution process and it can be computed locally for the selected regions and subdomains, also using parallel computing. [ABSTRACT FROM AUTHOR]

• The added degree of freedom method (ADM) is proposed to solve the absorbing boundary problem. • The principle of a mass damper is employed in the ADM to achieve the attenuation of stress waves. • The ADM can absorb one-dimensional and two-dimensional elastic waves across a broad range of frequencies. Absorbing boundaries are essential in engineering simulations, especially for elastodynamic problems, to ensure accuracy, reliability and numerical stability. In this paper, the added degree of freedom method (ADM) is developed as a novel approach to address absorbing boundary challenges in finite element analysis. In ADM, additional degrees of freedom (DOFs) are introduced within the absorbing domain to attenuate outgoing elastic waves. For the proposed ADM to implement the absorbing boundary, the stiffness and mass properties of both the added DOFs and conventional DOFs within the absorbing domain are adjusted. This adjustment aims to reduce the propagation speed of elastic waves within the medium, and to prolong the duration of interaction between elastic waves and the surrounding medium. Consequently, the vibrations of nodes can be effectively attenuated by applying damping forces to the added DOFs. The numerical results show that the ADM has the ability to absorb one-dimensional and two-dimensional elastic waves across a broad range of frequencies. Therefore, the proposed ADM offers an innovative solution for modeling absorbing boundaries in various scientific and engineering applications, addressing the challenges of simulating wave propagation within finite computational domains. [ABSTRACT FROM AUTHOR]

Abstract In this paper, a coupled peridynamic-finite element method for modeling mechanical behavior and damage growth in materials is developed. Peridynamics is a nonlocal continuum model which, in contrast to other continuum models, uses integration instead of spatial derivations in its governing equations. Utilizing integration instead of derivatives is advantageous in modeling fracture since the governing equations remain valid even after the initiation or growth of discontinuities. Although peridynamics can capture material fracture effectively, however due to its nonlocal formulation peridynamics is computationally expensive. To reduce the computational costs, we propose to couple peridynamics with finite elements and use peridynamics only in small zones where higher accuracy is needed. The main challenge in developing such a coupling method is to eliminate the artifacts introduced by the interface of the two subdomains. One of the main issues is spurious wave reflections which occurs because high frequency waves traveling from peridynamic region cannot enter the finite element zone and spuriously reflect back into the peridynamic zone. This will lead to an increase in the energy of the peridynamic zone and will drastically reduce the computational accuracy. The main feature of the proposed method is eliminating the spurious wave reflections such that the coupled method is as accurate as the pure peridynamics. Highlights • A coupled peridynamic-finite element method is developed to model dynamic fracture. • Spurious wave reflections are effectively suppressed. • Arlequin method is used to impose the mechanical compatibility between finite element and peridynamic sub regions. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we discuss the time-dependent generalized membrane shell model, which has not been addressed numerically in literature. We show that the solution of this model exists and is unique. We first provide a numerical method for the time-dependent generalized membrane shell. Concretely, we semi-discretize the space variable and fully discretize the problem using time discretization by the Newmark scheme. The corresponding numerical analyses of existence, uniqueness, stability and convergence with a priori error estimates are given. Finally, we present numerical experiments with a portion of the conical shell and a portion of the hyperbolic shell to verify theoretical convergence results and demonstrate the effectiveness of the numerical scheme. Highlights • Existence and uniqueness of the time-dependent generalized membrane shell. • Newmark scheme for the time-dependent generalized membrane shell. • Analyses of existence, uniqueness, stability, convergence and a priori error estimate for elastodynamic problem. [ABSTRACT FROM AUTHOR]

Abstract We propose and analyze a high-order Discontinuous Galerkin Finite Element Method for the approximate solution of wave propagation problems modeled by the elastodynamics equations on computational meshes made by polygonal and polyhedral elements. We analyze the well posedness of the resulting formulation, prove h p -version error a-priori estimates, and present a dispersion analysis, showing that polygonal meshes behave as classical simplicial/quadrilateral grids in terms of dispersion properties. The theoretical estimates are confirmed through various two-dimensional numerical verifications. Highlights • A new polygonal discontinuous Galerkin method is introduced for elastodynamics. • Stability and hp-version error estimates in a suitable energy norm are proved. • A dispersion analysis is carried out. • Dispersion errors are compared with those of classical spectral element methods. • The application of the method to benchmarks and realistic case studies is presented. [ABSTRACT FROM AUTHOR]

A polygonal scaled boundary finite element method (SBFEM) is proposed for linear elastodynamics in two dimensions. The domain is divided into non-overlapping polygonal elements, and the scaled boundary finite element approach is employed over each polygon. The advantages are that an arbitrary interpolation order can be used on each polygon and we can choose the optimal element order on each edge individually. Moreover, the SBFEM simplifies the numerical integration over the polygons when compared to conventional approaches. The dynamic stiffness matrix is computed by employing the continued fraction expansion. The influence of the shape of the polygon and the element order on the boundary of each polygon are studied. The robustness and the accuracy of the approach are demonstrated with numerical examples. [ABSTRACT FROM AUTHOR]

A new nodal mixed finite element is proposed for the simulation of linear elastodynamics and wave propagation problems in time domain. Our method is based on equal-order interpolation discrete spaces for both the velocity (or displacement) and stress (or strain) tensor variables. The mixed form is derived using either the velocity/stress or velocity/strain pair of unknowns, the latter being instrumental in extensions of the method to nonlinear mechanics. The proposed approach works equally well on hexahedral or tetrahedral grids and, for this reason, it is suitable for time-domain engineering applications in complex geometry. The peculiarity of the proposed approach is the use of the rate form of the stress update equation, which yields a set of governing equations with the structure of a non-dissipative space/time Friedrichs’ system. We complement standard traction boundary conditions for the stress with strongly and weakly enforced boundary conditions for the velocity (or displacement). Weakly enforced boundary conditions are particularly suitable when considering complex geometrical shapes, because they do not require dedicated data structures for the imposition of the boundary degrees of freedom, but, rather, they utilize the structure of the variational formulation. We also show how the framework of weakly enforced boundary conditions can be used to develop variational forms for multi-domain simulations of heterogeneous media. A complete analysis including stability and convergence proofs is included, in the case of a space–time variational approach. A series of computational tests are used to demonstrate and verify the performance of the proposed approach. [ABSTRACT FROM AUTHOR]

A new algorithm based on the convolution finite element method (CFEM) is proposed for the nonlinear wave propagation in elastic media. The formulation is developed in the context of the total Lagrangian framework, encompassing contributions due to both geometrical and material nonlinearities. As a basis, a counterpart of equations of motion – namely, the alternative field equations – is first established. The satisfaction of the alternative field equations is then realized in a weak sense. Next, the Newton–Raphson procedure and the consistent tangential matrix are applied to the weak formulation, where the CFEM is used as the linear solver in each iteration. Finally, several examples are carried out to examine the theoretical aspects and the feasibility of the proposed algorithm. In particular, problems of free vibration of Neo-Hookean and Saint Venant–Kirchhoff plates are explored. Also, a cantilever beam of the Neo-Hookean material is simulated for the case of forced vibrations. Conspicuously, in contrast to the existing time-step methods with finite order of accuracy, the new solution procedure obtains the accurate solution when the time-step size is increased. [ABSTRACT FROM AUTHOR]

In this work we present stabilized finite element methods for the mixed velocity–stress elasticity equations and for its irreducible velocity form. This is done both for the time and frequency domains, the latter being obtained by assuming a harmonic behavior in time. Stabilization methods that belong to the computational framework of the Variational Multi-Scale formulation are used. It is shown that the adequate selection of the algorithmic parameters on which the formulation depends allows one to switch from the primal to the dual functional framework. The performance of the method is tested through several numerical examples, one of which includes a convergence study. [ABSTRACT FROM AUTHOR]

We propose a stabilization method for linear tetrahedral finite elements, suitable for the implicit time integration of the equations of nearly and fully incompressible nonlinear elastodynamics. In particular, we derive and discuss a generalized framework for stabilization and implicit time integration that can comprehensively be applied to the class of all isotropic hyperelastic models. In this sense the presented development can be considered an important extension and complement to the stabilization approach proposed by the authors in previous work, which was instead focused on explicit time integration and simple neo-Hookean models for nearly-incompressible elasticity. With the goal of computational efficiency, we also present a two-step block Gauss–Seidel strategy for the time update of displacements, velocities and pressures. Specifically, a mixed system of equations for the velocity and pressure is updated implicitly in a first stage, and the displacements are updated explicitly in a second stage. The proposed mixed formulation is then embedded in Newton-type strategies for the nonlinear solution of the equations of motion. Various implicit time integration strategies are considered, and, particularly, we focus on high-frequency dissipation time integrators, which are preferable in transient mechanics applications. An extensive set of numerical computations with linear tetrahedral elements is presented to demonstrate the performance of the proposed approach. [ABSTRACT FROM AUTHOR]

A new finite element scheme is proposed to analyze the elastodynamics of materials having interactions between electrical and mechanical fields. Based on coupled constitutive equations and the alternative field equations, a new form of weighted residual in terms of the convolution product is established. Next, the Galerkin formulation is utilized with a particular form of spatial–temporal shape functions. High precision for arbitrary time intervals can be attained by reducing the integral forms to a set of algebraic equations. To show the accuracy of the proposed method, several examples, including elastodynamics of 1d and 2d piezoelectric materials under various initial and boundary conditions are presented. The present contribution introduces a new solution procedure for the analysis and design of active materials. [ABSTRACT FROM AUTHOR]

The aim of this paper is the design of a new one-step implicit and thermodynamically consistent Energy–Momentum (EM) preserving time integration scheme for the simulation of thermo-electro-elastic processes undergoing large deformations. The time integration scheme takes advantage of the notion of polyconvexity and of a new tensor cross product algebra. These two ingredients are shown to be crucial for the design of so-called discrete derivatives fundamental for the calculation of the second Piola–Kirchhoff stress tensor, the entropy and the electric field. In particular, the exploitation of polyconvexity and the tensor cross product, enable the derivation of comparatively simple formulas for the discrete derivatives. This is in sharp contrast to much more elaborate discrete derivatives which are one of the main downsides of classical EM time integration schemes. The newly proposed scheme inherits the advantages of EM schemes recently published in the context of thermo-elasticity and electro-mechanics, whilst extending to the more generic case of nonlinear thermo-electro-mechanics. Furthermore, the manuscript delves into suitable convexity/concavity restrictions that thermo-electro-mechanical strain energy functions must comply with in order to yield physically and mathematically admissible solutions. Finally, a series of numerical examples will be presented in order to demonstrate robustness and numerical stability properties of the new EM scheme. [ABSTRACT FROM AUTHOR]

We present a class of high-order finite element methods that can conserve the linear and angular momenta as well as the energy for the equations of linear elastodynamics. These methods are devised by exploiting and preserving the Hamiltonian structure of the equations of linear elastodynamics. We show that several mixed finite element, discontinuous Galerkin, and hybridizable discontinuous Galerkin (HDG) methods belong to this class. We discretize the semidiscrete Hamiltonian system in time by using a symplectic integrator in order to ensure the symplectic properties of the resulting methods, which are called symplectic Hamiltonian finite element methods. For a particular semidiscrete HDG method, we obtain optimal error estimates and present, for the symplectic Hamiltonian HDG method, numerical experiments that confirm its optimal orders of convergence for all variables as well as its conservation properties. • The first Hamiltonian structure-preserving HDG method for linear elastodynamics. • A complete analysis of energy-conserving Mixed, DG and HDG methods. • The linear and angular momenta, and the total energy remain constant in time. • HDG methods for linear elastodynamics with optimal error estimates. [ABSTRACT FROM AUTHOR]

The aim of this paper is the design a new one-step implicit and thermodynamically consistent Energy–Momentum (EM) preserving time integration scheme for the simulation of thermo-elastic processes undergoing large deformations and temperature fields. Following Bonet et al. (2020), we consider well-posed constitutive models for the entire range of deformations and temperature. In that regard, the consideration of polyconvexity inspired constitutive models and a new tensor cross product algebra are shown to be crucial in order to derive the so-called discrete derivatives, fundamental for the construction of the algorithmic derived variables, namely the second Piola–Kirchoff stress tensor and the entropy (or the absolute temperature). The proposed scheme inherits the advantages of the EM scheme recently published by Franke et al. (2018), whilst resulting in a simpler scheme from the implementation standpoint. A series of numerical examples will be presented in order to demonstrate the robustness and applicability of the new EM scheme. Although the examples presented will make use of a temperature-based version of the EM scheme (using the Helmholtz free energy as the thermodynamical potential and the temperature as the thermodynamical state variable), we also include in an Appendix an entropy-based analogue EM scheme (using the internal energy as the thermodynamical potential and the entropy as the thermodynamical state variable). • A new one-step implicit EM time integration scheme for nonlinear thermo-elasticity. • Four discrete derivatives are included in the algorithm. • Algebraic simplification in discrete derivatives with respect to existing algorithms. • Proof of thermodynamic consistency and of second order accuracy with respect to time. • Numerical examples demonstrate the robustness and applicability of the new EM scheme. [ABSTRACT FROM AUTHOR]

In this paper a new collocation technique for constructing time-dependent absorbing boundary conditions (ABCs) applicable to elastic wave motion is devised. The approach makes use of plane waves which satisfy the governing equations of motion to construct the absorbing boundary conditions. The plane waves are adjusted so that they can cope with the satisfaction of radiation boundary conditions. The proposed technique offers some advantages and exhibits the following features: it is easy to implement; its approximation scheme is local in space and time and thus it does not deal with any routine schemes such as Fourier and Laplace transform, making the method computationally less demanding; as the employed basis functions used to construct the absorbing boundary condition are residual-free, it requires neither any differential operator (to approximate the wave dispersion relation), nor any auxiliary variables; it constructs Dirichlet-type ABCs and hence no derivatives of the field variables are required for the imposition of radiation conditions. In this study, we apply the proposed technique to the solution procedure of a collocation approach based on the finite point method which proceeds in time by an explicit velocity-Verlet algorithm. It contributes to developing a consistent meshless framework for the solution of unbounded elastodynamic problems in time domain. We also apply the proposed method to a standard finite element solver. The performance of the method in solution of some 2D examples is examined. We shall show that the method exhibits appropriate results, conserves the energy almost exactly, and it performs stably in time even in the case of long-term computations. • Solution of transient elastic wave propagation problems in unbounded domains. • Proposing a new absorbing boundary condition local in space and time for elastodynamic problems. • Satisfaction of radiation conditions by imposing Dirichlet-type conditions in time-domain. • The meshless finite point method and the finite element method for unbounded problem domains. • Comparing the performance of the proposed approach with the conventional 1st order ABC method. [ABSTRACT FROM AUTHOR]