Search

In this paper, we are concerned with a class of second-order hemivariational inequalities involving history-dependent operators. For the problem, we first derive a semidiscrete scheme by implicit Euler formula and prove its unique solvability. The existence and uniqueness of a solution to the inequality problem is given by Rothe method. As the core part of the paper, we propose a two-step semidiscrete approximation for the problem, provide its unique solvability and obtain its second-order error estimates. The two-step scheme is more accurate than the standard implicit Euler scheme. Finally, we apply the results to a dynamic frictionless contact problem with long memory. [ABSTRACT FROM AUTHOR]

This paper aims to develop an efficient numerical scheme for simulating solid-state dewetting with strongly anisotropic surface energies in two dimensions. The governing equation is a sixth-order, highly nonlinear geometric partial differential equation, which makes it quite challenging to design an efficient numerical scheme. To tackle this problem, we first introduce an appropriate tangent velocity of the interface curve which could help mesh points equally distribute along the curve, then we reformulate the governing equation in terms of the tangent angle θ and the length L of the interface curve with a release of the stiffness brought by surface tension. To further reduce the numerical stability constraint from the high-order PDE, we propose a mixed finite element method for solving the reformulated θ -L equations. Numerical results are provided to demonstrate that the θ -L approach is not only efficient and accurate, but also has the mesh equidistribution property with an improved numerical stability. [ABSTRACT FROM AUTHOR]

Most building materials can be characterized as quasi-brittle composites with a cementitious matrix, reinforced by some stiffening particles or elements. Their massive exploitation motivates the development of numerical modelling and simulation of behaviour of such material class under mechanical, thermal, etc. loads, including the evaluation of the risk of initiation and development of micro- and macro-fracture. This paper demonstrates the possibility of certain deterministic prediction, applying the dynamical approach using the Kelvin viscoelastic model and cohesive interface properties. The existence and convergence results rely on the semilinear computational scheme coming from the method of discretization in time, using several types of Rothe sequences, coupled with the extended finite element method (XFEM) for practical calculations. Numerical examples refer to cementitious samples reinforced by short steel fibres, with increasing number of applications as constructive parts in civil engineering. [ABSTRACT FROM AUTHOR]

This paper investigates a novel parallel technique based on the spectral deferred correction (SDC) method and a compensation step for solving first-order evolution problems, and we call it para-SDC method for convenience. The standard SDC method is used in parallel with a rough initial guess and a Picard integral equation with high precision initial condition is acted as a compensator. The goal of this paper is to show how these processes can be parallelized and how to improve the efficiency. During the SDC step an implicit or semi-implicit method can be used for stiff problems which is always time-consuming, therefore that’s why we do this procedure in parallel. Due to a better initial condition of parallel intervals after the SDC step, the goal of compensation step is to get a better approximation and also avoid of solving an implicit problem again. During the compensation step an explicit Picard scheme is taken based on the numerical integration with polynomial interpolation on Gauss Radau II nodes, which is almost no time consumption, obviously, that’s why we do this procedure in serial. The convergency analysis and the parallel efficiency of our method are also discussed. Several numerical experiments and an application for simulation Allen-Cahn equation are presented to show the accuracy, stability, convergence order and efficiency features of para-SDC method. [ABSTRACT FROM AUTHOR]

The Kirchhoff approximation is widely used to describe the scatter of elastodynamic waves. It simulates the scattered field as the convolution of the free-space Green's tensor with the geometrical elastodynamics approximation to the total field on the scatterer surface and, therefore, cannot be used to describe nongeometrical phenomena, such as head waves. The aim of this paper is to demonstrate that an alternative approximation, the convolution of the far-field asymptotics of the Lamb's Green's tensor with incident surface tractions, has no such limitation. This is done by simulating the scatter of a critical Gaussian beam of transverse motions from an infinite plane. The results are of interest in ultrasonic nondestructive testing. [ABSTRACT FROM AUTHOR]

The paper considers the time integration of frictionless dynamical contact problems between viscoelastic bodies in the frame of the Signorini condition. Among the numerical integrators, interest focuses on the classical Newmark method, the improved energy dissipative version due to Kane et al., and the contact-stabilized Newmark method recently suggested by Deuflhard et al. In the absence of contact, any such variant is equivalent to the Störmer–Verlet scheme, which is well-known to have consistency order 2. In the presence of contact, however, the classical approach to discretization errors would not show consistency at all because of the discontinuity at the contact. Surprisingly, the question of consistency in the constrained situation has not been solved yet. The present paper fills this gap by means of a novel proof technique using specific norms based on earlier perturbation results due to the authors. The corresponding estimation of the local discretization error requires the bounded total variation of the solution. The results have consequences for the construction of an adaptive timestep control, which will be worked out subsequently in a forthcoming paper. [ABSTRACT FROM AUTHOR]

Analytical and numerical studies of multi-degree-of-freedom (MDOF) nonlinear stochastic or deterministic dynamic systems have long been a technical challenge. This paper presents a highly-efficient method for determining the stationary probability density functions (PDFs) of MDOF nonlinear systems subjected to both additive and multiplicative Gaussian white noises. The proposed method takes advantages of the sufficient conditions of the reduced Fokker-Planck-Kolmogorov (FPK) equation when constructing the trial solution. The assumed solution consists of the analytically constructed trial solutions satisfying the sufficient conditions and an exponential polynomial of the state variables, and delivers a high accuracy of the solution because the analytically constructed trial solutions capture the main characteristics of the nonlinear system. We also make use of the concept from the data-science and propose a symbolic integration over a hypercube to replace the numerical integrations in higher-dimensional space, which has been regarded as the insurmountable difficulty in the classical method of weighted residuals or stochastic averaging for high-dimensional dynamic systems. Three illustrative examples of MDOF nonlinear systems are analyzed in detail. The accuracy of the numerical results is validated by comparison with the Monte Carlo simulation (MCS) or the available exact solution. Furthermore, we also show the substantial gain in the computational efficiency of the proposed method compared with the MCS. [ABSTRACT FROM AUTHOR]

Piezoelectric materials have become one of the most promising candidates for a new class of smart structures which are widely used as sensors and actuators in control systems due to their superior electromechanical performance and high‐efficiency in electro‐mechanical energy conversion. In recent years, although there are many papers of theoretical modeling of generalized piezoelectric thermoelasticity that considered the piezoelectric‐thermoelastic coupling of piezoelectric structures serving in the extreme fast heating environment (e.g. ultra‐fast laser), the strain relaxation effect has not been considered yet. This work aims to investigate the generalized piezoelectricity problem with strain rate with the aid of extended thermodynamic principles, and corresponding generalized variational theorem is also strictly formulated to provide a complete rationale for numerical method by using semi‐inverse method. For numerical evaluation, transient thermo‐electromechanical responses of a one‐dimensional semi‐infinite piezoelectric medium with one end subjecting to a sudden thermal loading are considered and solved by applying Laplace transformation methods. Then the transient responses of temperature, displacement, strain, stress and electric potential are obtained and illustrated graphically, and the influences of strain relaxation parameters on them are also analyzed and discussed. [ABSTRACT FROM AUTHOR]

The vibroimpact systems with bilateral barriers are often encountered in practice. However, the dynamics of the vibroimpact system with bilateral barriers is full of challenges. Few closed-form solutions were obtained. In this paper, we propose a novel method for random vibration analysis of single-degree-of-freedom (SDOF) vibroimpact systems with bilateral barriers under Gaussian white noise excitations. A periodic approximate transformation is employed to convert the equations of the motion to a continuous form. The probabilistic description of the system is subsequently defined through the corresponding Fokker-Planck-Kolmogorov (FPK) equation. The closed-form stationary probability density function (PDF) of the response is obtained by solving the reduced FPK equation and using the proposed iterative method of weighted residue together with the concepts of the circulatory probability flow and the potential probability flow. Finally, the versatility of the proposed approach is demonstrated by its application to two typical examples. Note that the solution obtained by using the proposed method can be used as the benchmark to examine the accuracy of approximate solutions obtained by other methods. [ABSTRACT FROM AUTHOR]

This paper deals with a fully implicit time discretization scheme with variable time-step for a nonlinear system modelling phase transition and mechanical deformations in shape memory alloys. The model is studied in the non-stationary case and accounts for local microscopic interactions between the phases introducing the gradients of the phase parameters. The resulting initial-boundary value problem has already been studied by the author who proved existence, uniqueness and continuous dependence on data for a suitable weak solution along some regularity results. A careful and detailed investigation of the variable time-step discretization is the goal of this paper. Thus, we deduce some estimates for the discretization error. These estimates depend only on data, impose no constraints between consecutive time-steps and show an optimal order of convergence. Finally, we prove another regularity result for the solution under stronger regularity assumptions on data. [ABSTRACT FROM AUTHOR]

The paper considers a slightly improved variant of the contact-stabilized Newmark method by Deuflhard et al., which provides a spatio-temporal numerical integration of dynamical contact problems between viscoelastic bodies in the frame of the Signorini condition. Up to now, the question of consistency in the case of contact constraints has been discussed for time integrators in function space under the assumption of bounded total variation of the solution. Here, interest focuses on the consistency error of the Newmark scheme in physical energy norm after discretization both in time and in space. The resulting estimate for the local discretization error allows to prove global convergence of the Newmark scheme under an additional assumption on the active contact boundaries. [ABSTRACT FROM AUTHOR]

In this paper, we consider the modified epidemiological model for computer viruses (SAIR) proposed by Piqueira and Araujo (Appl Math Comput 2(213):355-360, ). The multi-step homotopy analysis method (MHAM) is employed to compute an approximation to the solution of the model of fractional order. The fractional derivatives are described in the Caputo sense. Figurative comparisons between the MHAM and the classical fourth-order Runge-Kutta method reveal that this method is very effective. The solutions obtained are also presented graphically. [ABSTRACT FROM AUTHOR]

The discretization size is limited by the sampling theorem, and the limit is one half of the wavelength of the highest frequency of the problem. However, one half of the wavelength is an ideal value. In general, the discretization size that can ensure the accuracy of the simulation is much smaller than this value in the traditional finite element method. The possible reason of this phenomenon is analyzed in this paper, and an efficient method is given to improve the simulation accuracy. [ABSTRACT FROM AUTHOR]

The goal of this paper is to derive a structure preserving integrator for geometrically exact beam dynamics, by using a Lie group variational integrator. Both spatial and temporal discretization are implemented in a geometry preserving manner. The resulting scheme preserves both the discrete momentum maps and symplectic structures, and exhibits almost-perfect energy conservation. Comparisons with existing numerical schemes are provided and the convergence behavior is analyzed numerically. [ABSTRACT FROM AUTHOR]

We propose a new Cahn–Hilliard phase field model coupled to incompressible viscoelasticity at large strains, obtained from a diffuse interface mixture model and formulated in the Eulerian configuration. A new kind of diffusive regularization, of Allen–Cahn type, is introduced in the transport equation for the deformation gradient, together with a regularizing interface term depending on the gradient of the deformation gradient in the free energy density of the system. The designed regularization preserves the dissipative structure of the equations. We obtain the global existence of a weak solution in three space dimensions and for generic nonlinear elastic energy densities with polynomial growth, comprising the relevant cases of polyconvex Mooney–Rivlin and Ogden elastic energies. Also, our analysis considers elastic free energy densities which depend on the phase field variable and which can possibly degenerate for some values of the phase field variable. We also propose two kinds of unconditionally energy stable finite element approximations of the model, based on convex splitting ideas and on the use of a scalar auxiliary variable respectively, proving the existence and stability of discrete solutions. We finally report numerical results for different test cases with shape memory alloy type free energy, showing the interplay between phase separation and finite elasticity in determining the topology of stationary states with pure phases characterized by different elastic properties. [ABSTRACT FROM AUTHOR]

This paper provides the formulation used for studing the cold and hot separating stages of a multistage launch vehicle. Monte Carlo simulation is employed to account for the off nominal design parameters of the bodies undergoing separation to evaluate the risk of failure for the separation event. All disturbances, effect of dynamic unbalance, residual thrust, separation disturbance caused by the separation mechanism and misalignment in cold and hot separation are analyzed to find out nonoccurrence of collision between the separation bodies. The results indicate that the current design satisfies the separation requirements. [ABSTRACT FROM AUTHOR]

There has been a considerable number of recent discussions on applications of different types of dynamic derivatives. The dynamic derivatives not only play a central role in time scales theory and methods, but also provide a remarkable way of approximations in mathematical modelling and computational applications. In this paper, we will explore connections between existing dynamic and conventional derivatives over the time scales used. We will show that while first order dynamic derivatives provide acceptable accuracies in approximating conventional derivatives, second order dynamic derivatives are in general inconsistent with conventional derivatives. It is evident that, before incorporating structures of particular time scales used into the dynamic derivative formulae properly, one needs to be prudential when bridging discrepancies between the differential and difference equations modelling similar applications by higher order dynamic derivatives. Numerical experiments are given to further illustrate our conclusions. [ABSTRACT FROM AUTHOR]

A fully discrete finite element scheme is proposed to solve the dynamic viscoelastic contact problem with multi-contact zones and unilateral constraints. We develop the well-posedness and error analysis of the nonlinear numerical scheme under appropriate assumptions on the regularity of the solution. Moreover, we provide a projection-iteration algorithm for computation and show the convergence. Several numerical experiments are carried out to validate the theoretical results. [ABSTRACT FROM AUTHOR]

In this work, the initial-boundary value problem is considered for the dynamic Kirchhoff string equation u t ⁢ t - (α ⁢ (t) + β ⁢ ∫ - 1 1 u x 2 ⁢ d x ) ⁢ u x ⁢ x = f . Here α ⁢ (t) is a continuously differentiable function, α ⁢ (t) ≥ c 0 > 0 and 훽 is a positive constant. For solving this problem approximately, a symmetric three-layer semi-discrete scheme with respect to the temporal variable is applied, in which the value of a nonlinear term is taken at the middle point. This approach allows us to find numerical solutions per temporal steps by inverting the linear operators. In other words, applying this scheme, a system of linear ordinary differential equations is obtained. The local convergence of the scheme is proved. The results of numerical computations using this scheme for different test problems are given for which the Legendre–Galerkin spectral approximation is applied with respect to the spatial variable. [ABSTRACT FROM AUTHOR]

The problem of hydraulic fracture for the PKN model is considered within the framework presented recently by Linkov (Doklady Phys 56(8):436–438, 2011 ). The modified formulation is further enhanced by employing an improved regularized boundary condition near the crack tip. This increases solution accuracy especially for singular leak-off regimes. A new dependent variable having clear physical sense is introduced. A comprehensive analysis of numerical algorithms based on various dependent variables is provided. Comparison with know numerical results has been given. [ABSTRACT FROM AUTHOR]

The stress- strain time relationship Graph HT ht (1.1) proposed by Zener [[27]] gives rise to the simplest viscoelastic model that takes into account important features such as creep/recovery and stress relaxation phenomena [[24]]. Keywords: mixed finite elements; linear viscoelasticity; error estimates; 65N30; 65M12; 65M15; 74H15 EN mixed finite elements linear viscoelasticity error estimates 65N30 65M12 65M15 74H15 43 62 20 03/11/22 20220301 NES 220301 1 Introduction We aim to study the propagation of waves in linear structures composed of materials with purely elastic parts and viscoelastic parts. We also introduce the Hilbert spaceH( B div b , I i ):= { B I i b L SP 2 sp ( I i ): B div I i b B L b SP 2 sp ( I i )} and denote the corresponding norm HT ht Let B I n i b be the outward unit normal vector to I i . [Extracted from the article]

This review article revisits and outlines the perfectly matched layer (PML) method and its various formulations developed over the past 25 years for the numerical modeling and simulation of wave propagation in unbounded media. Based on the concept of complex coordinate stretching, an efficient mixed displacement-strain unsplit-field PML formulation for second-order (displacement-based) linear elastodynamic equations is then proposed for simulating the propagation and absorption of elastic waves in unbounded (infinite or semi-infinite) domains. Both time-harmonic (frequency-domain) and time-dependent (time-domain) PML formulations are derived for two- and three-dimensional linear elastodynamic problems. Through the introduction of only a few additional variables governed by low-order auxiliary differential equations, the resulting mixed time-domain PML formulation is second-order in time, thereby allowing the use of standard time integration schemes commonly employed in computational structural dynamics and thus facilitating the incorporation into existing displacement-based finite element codes. For computational efficiency, the proposed time-domain PML formulation is implemented using a hybrid approach that couples a mixed (displacement-strain) formulation for the PML region with a classical (displacement-based) formulation for the physical domain of interest, using a standard Galerkin finite element method (FEM) for spatial discretization and a Newmark time scheme coupled with a finite difference (Crank-Nicolson) time scheme for time sampling. Numerical experiments show the performances of the PML method in terms of accuracy, efficiency and stability for two-dimensional linear elastodynamic problems in single- and multi-layer isotropic homogeneous elastic media. [ABSTRACT FROM AUTHOR]

This research deals with the study of thermoelastic damping in transversely isotropic thin circular Kirchhoff–Love plate. Mathematical model is formed for time-harmonic displacement and temperature fields due to the GN theory of thermoelasticity of type III. The model is solved to obtain the expressions for thermoelastic damping, displacement, and temperature fields for circumferential surface wave modes for simply supported and clamped plate resonators. The numerical results for both the simply supported and clamped boundary conditions for an axisymmetric circular plate are illustrated. It is observed that the damping of vibrations is considerably influenced by surface wave modes. The thickness of the plate also has major effects on the vibrations of resonators. The computer-simulated results to illustrate the effect of circumferential surface wave modes on thermoelastic damping for the GN III theory of thermoelasticity have been depicted graphically. [ABSTRACT FROM AUTHOR]

We present an arbitrary high-order local discontinuous Galerkin (LDG) method with alternating fluxes for solving linear elastodynamics problems in isotropic media. Both the semi-discrete analysis and fully discrete analysis for a leap-frog LDG method are given to show that the proposed method simultaneously enjoys the energy conserving property and optimal convergence rates in both the displacement and stress, when the tensor product polynomials of the degree k are used on Cartesian meshes. Numerical experiments demonstrate that the proposed method has several advantages including the exact energy conservation, slow-growing errors in long time simulation, and subtle dependence on the first Lamé parameter λ . [ABSTRACT FROM AUTHOR]

In this paper the authors aspire to obtain the approximate analytical solution of Modified Burgers Equation with newly defined conformable derivative by employing homotopy analysis method (HAM).

We introduce and analyze a new mixed finite element method with reduced symmetry for the standard linear model in viscoelasticity. Following a previous approach employed for linear elastodynamics, the present problem is formulated as a second-order hyperbolic partial differential equation in which, after using the motion equation to eliminate the displacement unknown, the stress tensor remains as the main variable to be found. The resulting variational formulation is shown to be well-posed, and a class of H (div) -conforming semi-discrete schemes is proved to be convergent. Then, we use the Newmark trapezoidal rule to obtain an associated fully discrete scheme, whose main convergence results are also established. Finally, numerical examples illustrating the performance of the method are reported. [ABSTRACT FROM AUTHOR]

A subspace expanding technique (SET) is proposed to efficiently discover and find all zeros of nonlinear functions in multi-degree-of-freedom (MDOF) engineering systems by discretizing the space into smaller subdomains, which are called cells. The covering set of the cells is identified by parallel calculations with the root bracketing method. The covering set can be found first in a low-dimensional subspace, and then gradually extended to higher dimensional spaces with the introduction of more equations and variables into the calculations. The results show that the proposed SET is highly-efficient for finding zeros in high-dimensional spaces. The subdivision technique of the cell mapping method is further used to refine the covering set, and the obtained numerical results of zeros are accurate. Three examples are further carried out to verify the applicability of the proposed method, and very good results are achieved. It is believed that the proposed method will significantly enhance the ability to study the stability, bifurcation, and optimization problems in complex MDOF nonlinear dynamic systems. [ABSTRACT FROM AUTHOR]

We study a fourth-order, singular, nonlinear PDE model for surface relaxation. A weak solution for the model is defined using an inequality formulation. A numerical scheme based on semi-implicit time stepping, mixed finite elements and regularization is proposed to approximate the PDE model. We investigate the convergence of the scheme with respect to the discretization and the regularization parameters. Finally, formal arguments show that the model can be viewed as a gradient flow with respect to an appropriate Riemannian metric. [ABSTRACT FROM AUTHOR]

The free vibration analysis of a rotating cylindrical shell with an analytical method is investigated. The shell is considered as a sandwich structure, where the middle layer is a functionally graded material (FGM) shell, and it is surrounded by two piezoelectric layers. Considering piezoelectric materials to be functionally graded (FG), the material properties vary along the thickness direction as one innovation of this study. Applying the first-order shear deformation theory (FSDT), the equations of motion of this electromechanical system are derived as the partial differential equations (PDEs) using Hamilton's principle. Then, the Galerkin procedure is used to discretize the governing equations, and the present results are compared with the previously published results for both isotropic and FGM shells to verify the analytical method. Finally, the effects of FGM and functionally graded piezoelectric material (FGPM) properties as well as the thickness ratio and the axial and circumferential wave numbers on the natural frequencies are studied. Moreover, the Campbell diagram is plotted and discussed through the governing equations. The present results show that increasing the non-homogeneous index of the FGM decreases the natural frequencies on the contrary of the effect of non-homogeneous index of the FGPM. [ABSTRACT FROM AUTHOR]

We propose an implicit Newmark method for the time integration of the pressure-stress formulation of a fluid-structure interaction problem. The space Galerkin discretization is based on the Arnold-Falk-Winther mixed finite element method with weak symmetry in the solid and the usual Lagrange finite element method in the acoustic medium. We prove that the resulting fully discrete scheme is well-posed and uniformly stable with respect to the discretization parameters and Poisson ratio, and we provide asymptotic error estimates. Finally, we present numerical tests to confirm the asymptotic error estimates predicted by the theory. [ABSTRACT FROM AUTHOR]