Elastodynamic Problems Research Articles

Mathematical modelling of acoustic radiation force in transient shear wave elastography in the heart

The aim of this work is to provide a mathematical model and analysis of the excitation and the resulting shear wave propagation in acoustic radiation force (ARF)-based shear wave cardiac elastography. Our approach is based on asymptotic analysis; more precisely, it consists in considering a family of problems, parametrised by a small parameter inversely proportional to the excitation frequency of the probes, the viscosity and the velocity of pressure wave propagation. We derive a simplified model for the expression of the ARF by investigating the limit behaviour of the solution when the small parameter goes to zero. By formal asymptotic analysis – an asymptotic expansion of the solution is used – and energy analysis of the nonlinear elastodynamic problem, we show that the leading-order term of the expansion is solution of the underlying, incompressible, nonlinear cardiac mechanics. Subsequently, two corrector terms are derived. The first is a fast-oscillating pressure wave generated by the probes, solution of a Helmholtz equation at every time. The second corrector term consists in an elastic field with prescribed divergence, having a function of the first corrector as a source term. This field corresponds to the shear acoustic wave induced by the ARF. We also confirm that, in cardiac mechanics, the presence of viscosity in the model is essential to derive an expression of the shear wave propagation from the ARF, and that this phenomenon is related to the nonlinearity of the partial differential equation.

Diffusive subsurface propagation of Rayleigh mode induced by embedded ultrasonic waveguide

Surface degradation in the wood surface has generated unwanted variation in the ultrasound and reduced the efficacy of ultrasonic based non-destructive evaluation (NDE). This study proposes a novel alternative by inserting a waveguide into the wooden medium to eliminate the impact of surface variation on the characteristics of the ultrasonic propagation. This technique creates a small circular insertion of 2 cm depth with 5 mm in diameter, which has been accepted by many utility inspection firms for structural health monitoring (SHM) of wooden utility grids. This work develops a closed-form analytical solution of the elastodynamic problem using the proposed embedded waveguide technique. The resulted solution based on the Navier's formulation discovered two interfering wave modes embedded in the displacement field. These two modes termed fast and slow induce diffusive displacement propagation as a function of the Poisson's ratio of the medium. The analytical result also shows a greater elastic energy penetration as a function of the insertion depth. Resulted wave characteristics are different from the traditional contact-based Rayleigh excitation. A developed numerical model using the semi-explicit differential-algebraic equation (DAE) technique with both steady-state and transient load conditions is demonstrated. The simulated displacement fields are analyzed and compared with the analytical result.

Stable IMEX schemes for a Nitsche-based approximation of elastodynamic contact problems. Selective mass scaling interpretation

We introduce some IMEX schemes (implicit-explicit schemes with an implicit term being linear) for approximating elastodynamic contact problems when the contact condition is taken into account with a Nitsche method. We develop a theoretical and numerical study of the properties of the schemes, especially in terms of stability, provide some numerical comparisons with standard explicit and implicit scheme and propose some improvements to obtain a more reliable approximation of motion for large time steps. We also show how selective mass scaling techniques can be interpreted as IMEX schemes.

New Class of Homogeneous Solutions for Plane Elastodynamic Problems

The paper presents a new type of functionally invariant Smirnov–Sobolev solutions for the wave equation, which can be used for solving many homogeneous problems of elastodynamics. The derived solution has a unique property: the double preimage of Laplace and Fourier transforms of this function with a new argument and coefficient coincides with the function itself. Thus, this property allows us to find an inversion formula for double integral transformations. The method for obtaining this formula is demonstrated by solving the Lamb problem for a half-plane under the transition from the image to the original.

Finite element method and boundary element method iterative coupling algorithm for 2-D elastodynamic analysis

The coupling algorithm of the finite element method (FEM) and boundary element method (BEM) can make maximal use of both methods’ advantages. However, such coupling will reduce the computational efficiency because the systems’ degrees of freedom will increase sharply. Thus, a new coupling algorithm that achieves accuracy and computational efficiency is necessary. This study proposes the Newmark-based precise integration FEM (NBPI-FEM) and analytical-based time domain BEM (ABTD-BEM) coupling algorithm. In this coupling algorithm, the governing equation of the FEM is solved by Newmark-based precise integration, and the governing equation of the BEM is solved by the analytical method. First, the procedures of NBPI-FEM and ABTD-BEM are given. The coupling strategy is then provided, and the relationship between the iteration numbers and the relaxation parameter is investigated. Finally, two illustrative examples—i.e., 1-D rod and a semi-infinite structure—are selected to verify the coupling algorithm proposed in the study. The results show that the numerical solutions agree well with the analytical solutions for the 1-D rod example and coincide with the numerical solutions calculated by FEM. Thus, NBPI-FEM and ABTD-BEM can be applied for solving elastodynamic problems with high accuracy and efficiency.

Harmonic Standing-Wave Excitations of Simply-Supported Isotropic Solid Elastic Circular Cylinders: Exact 3D Linear Elastodynamic Response

The vibration of a solid elastic cylinder is one of the classical applied problems of elastodynamics. Many fundamental forced-vibration problems involving solid elastic cylinders have not yet been studied or solved using the full three-dimensional (3D) theory of linear elasticity. One such problem is the steady-state forced-vibration response of a simply-supported isotropic solid elastic circular cylinder subjected to two-dimensional harmonic standing-wave excitations on its curved surface. In this paper, we exploit certain recently-obtained particular solutions to the Navier–Lamé equation and exact matrix algebra to construct an exact closed-form 3D elastodynamic solution to the problem. The method of solution is direct and demonstrates a general approach that can be applied to solve other similar forced-vibration problems involving elastic cylinders. The obtained analytical solution is then applied to a specific numerical example, where it is used to determine the frequency response of the displacement field in some low wave number excitation cases. In each case, the solution generates a series of resonances that are in exact correspondence with a subset of the natural frequencies of the simply-supported cylinder. The analytical solution is also used to compute the resonant mode shapes in some selected asymmetric excitation cases. The studied problem is of general interest both as an exactly-solvable 3D elastodynamics problem and as a benchmark forced-vibration problem involving a solid elastic cylinder.

Two-dimensional elastodynamic and free vibration analysis by the method of fundamental solutions

In this paper we will present a formulation for solving elastodynamic problems in 2D using the method of fundamental solutions (MFS). The governing equation for the displacement in the elastodynamic problem is expressed by the Navier equation with an additional non-homogeneous inertial term, which involves the second derivative of displacement with respect to time. The inertial term is approximated by the Houbolt finite difference formula. In each time step, the solution for the displacement is separated into a homogeneous solution and a particular solution. The homogeneous solution is constructed in terms of the fundamental solutions of the elastostatic problem. The particular solution corresponds to the non-homogeneous inertial term, which is obtained by approximating the inertial term with radial basis functions (RBFs). In addition, a formulation based on the MFS for free vibration analysis of 2D bodies is presented in this paper. By solving several numerical examples, we have demonstrated that the proposed methods are robust and accurate. The convergence of the results with respect to the number of collocation points, the number of internal points, the value of the RBF parameter, and the time step size is also studied.

Influence of soil non-homogeneity on the base shear force of piled structures subjected to harmonic seismic waves

This paper aims at analyzing the effects of soil non-homogeneity on the dynamic behavior of piled structures. In order to fulfil this objective a procedure based on a substructuring model, which includes soil-structure interaction effects, is used to compute the dynamic response of slender and non-slender structures supported on several pile group configurations embedded in four different linear-elastic Gibson soil profiles. A numerical model based on the integral formulation of the elastodynamic problem and Green’s functions for the layered half space is used for the purpose of determining the foundation dynamic response in terms of impedances and kinematic interaction factors. The results obtained in terms of structural base shear force for each one of the variable-with-depth soil profiles are compared to those computed for an equivalent homogeneous soil. A significant influence of the soil profile on the structural harmonic response to seismic waves is observed in terms of maximum shear force at the base of the structure. The homogeneous assumption when modelling depth-varying soil profiles yields, in many cases, non-conservative results when predicting the structural dynamic behavior.

On the efficiency of nested GMRES preconditioners for 3D acoustic and elastodynamic [formula omitted]-matrix accelerated Boundary Element Methods

This article is concerned with the derivation of fast Boundary Element Methods for 3D acoustic and elastodynamic problems. In particular, we are interested in the acceleration of Hierarchical matrix (H-matrix) based iterative solvers. While H-matrix representations allow to reduce the storage requirements and the cost of a matrix–vector product, the number of iterations for an iterative solver, as the frequency or the problem size increases, remains an issue.We consider an inner–outer preconditioning strategy, i.e., the preconditioner is applied through an iterative solver at the inner level. The preconditioner is defined as a H-matrix representation of the system matrix with a given accuracy. We investigate the influence of various parameters of the preconditioner, i.e., the H-matrix accuracy, the GMRES threshold and the maximum number of iterations of the inner solver. Different numerical results are presented to compare the efficiency of the preconditioner with respect to the unpreconditioned reference system. Finally, we propose a way to define the optimal setting for this preconditioner.

An adaptive cellular automata approach with the use of radial basis functions for the simulation of elastic wave propagation

A physics-based adaptive approach constructed on cellular automata is proposed for the analysis of two-dimensional (2D) elastodynamic initial boundary value problems. Although any 2D cell may be used in this approach, because of easy construction of arbitrary geometry, triangular plane cells are used as primary reference. The analysis of the problem includes two principal parts. In the first part, after discretization of the problem domain, a physics-based configuration considering the conservation of mass and the balance of momentum is developed as reference for cellular automata used in the next step. In the second part, radial basis functions (RBFs) are employed for an adaptive solution. For this purpose, after analysing the problem based on an arbitrary coarse mesh in the first step as reference, the problem domain may be discretized by iterative schemes, where RBFs indicate the lack and redundancy of computational points of the domain, from which some computational points may be added or eliminated for the next analysis. This process may be repeated to get the most appropriate mesh with the most accurate results. The principle of causality is taken into account in the proposed method, meaning that the unexcited parts of the domain are not in the solution domain. Similarly, those parts of the domain where the wave has already passed from would be deducted from the solution domain. The results of the proposed approach are compared with numerical and analytical results available in the literature. The comparison illustrates very good agreements between the results of the proposed approach and the results of those already reported in the literature.

Randomized residual‐based error estimators for the proper generalized decomposition approximation of parametrized problems

SummaryThis article introduces a novel error estimator for the proper generalized decomposition (PGD) approximation of parametrized equations. The estimator is intrinsically random: it builds on concentration inequalities of Gaussian maps and an adjoint problem with random right‐hand side, which we approximate using the PGD. The effectivity of this randomized error estimator can be arbitrarily close to unity with high probability, allowing the estimation of the error with respect to any user‐defined norm as well as the error in some quantity of interest. The performance of the error estimator is demonstrated and compared with some existing error estimators for the PGD for a parametrized time‐harmonic elastodynamics problem and the parametrized equations of linear elasticity with a high‐dimensional parameter space.

Local Dirichlet-type absorbing boundary conditions for transient elastic wave propagation problems

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.

A Mixed Stress/Displacement Approach Model of Homogeneous Shells for Elastodynamic Problems

This paper presents the development of a model of homogeneous, moderately thick shells for elastodynamic problems. The model is obtained by adapting and modifying SAM-H model (stress approach model of homogeneous shells) developed by Domínguez Alvarado and Díaz in (2018) for static problems. In the dynamic version of SAM-H presented herein, displacements and stresses are approximated by polynomials of the out-of-plane coordinate. The stress approximation coincides with the static version of SAM-H when dynamic effects are neglected. The generalized forces and displacements appearing in the approximations are the same as those involved in a classical, moderately thick shell model (CS model) but the stress approximation adopted herein is more complex: the 3D motion equations and the stress boundary conditions at the faces of the shell are verified. The generalized motion and constitutive equations of dynamic SAM-H model are obtained by applying a variant of Euler–Lagrange equation which includes pertinently Hellinger–Reissner functional. In the constitutive equations, Poisson’s effect of out-of-plane normal stresses on in-plane strains is not ignored; this is one important feature of SAM-H. To test the accuracy of dynamic SAM-H model, the following structures were considered: a hollow sphere and a catenoid. In each case, eigenfrequencies are first calculated and then a frequency analysis is performed applying a harmonic load. The results are compared to those of a CS model, MITC6 (mixed interpolation of tensorial components with 6 nodes per element) shell element calculations, and solid finite element computations. In the two problems, CS, MITC6, and dynamic SAM-H models yield accurate eigenfrequencies and eigenmodes. Nevertheless, the frequency analysis performed in each case showed that dynamic SAM-H provides much more accurate amplitudes of stresses and displacements than the CS model and the MITC6 shell finite element technique.

An application of the Burton-Miller method to the isogeometric boundary element method for 2D elastodynamic problems

An application of the Burton-Miller method to the isogeometric boundary element method for 2D elastodynamic problems

Accurate numerical solutions of 2-D elastodynamics problems using compact high-order stencils

A new approach for the increase in the order of accuracy of high-order numerical techniques used for 2-D time-dependent elasticity (structural dynamics and wave propagation) has been suggested on uniform square and rectangular meshes. It is based on the optimization of the coefficients of the corresponding semi-discrete stencil equations with respect to the local truncation error. It is shown that the second order of accuracy of the linear finite elements with 9-point stencils is optimal and cannot be improved. However, we have calculated new 9-point stencils with the second order of accuracy that yield more accurate results than those obtained by the linear finite elements. We have also developed new 25-point stencils (similar to those for the quadratic finite and isogeometric elements) with the optimal sixth order of accuracy for the non-diagonal mass matrix and with the optimal fourth order of accuracy for the diagonal mass matrix. The numerical results are in good agreement with the theoretical findings as well as they show a big increase in accuracy of the new stencils compared with those for the high-order finite elements. At the same number of degrees of freedom, the new approach yields significantly more accurate results than those obtained by the high-order (up to the eighth order) finite elements with the non-diagonal mass matrix. The numerical experiments also show that the new approach with 25-point stencils yields very accurate results for nearly incompressible materials with Poisson’s ratio 0.4999.

Singularity cancellation method for time-domain boundary element formulation of elastodynamics: A direct approach

In the implementation of time-domain boundary element method for elasto-dynamic problems, there are two types of singularities: the wave front singularity arising when the product of wave velocity and time is equal to the distance between the source point and the field point, and the spatial singularity arising when the source point coincides with the field point. In this paper, the singularity of the first type in the integrand is eliminated by an analytical integration over time, Cauchy principal value and Hadamard finite part integral. Four types of singularities with different orders appear in the integrand after analytical time integration. In order to accurately calculate the integral, in which the integrand is piecewise continuous, the integral domain is subdivided into several patches based on the relation between the product of wave velocity and time and the distance. In singular patches, the integrands are separated into a regular part and a singular part. The regular part can be computed by traditional numerical integration method such as Gaussian integration, while the singular part can be analytically integrated. Using the proposed method, the spatial singular integrals can be calculated directly. Numerical examples using various kinds of elements are presented to verify the proposed method.

Local well-posedness of the free-surface incompressible elastodynamics

We are concerned with the free boundary problem of three-dimensional incompressible neo-Hookean elastodynamics in physical vacuum, which describes the motion of neo-Hookean elastic waves in an incompressible material. Up to now, only local-in-time a priori estimates for the solutions of the problem has been established by Hao-Wang [16] through a geometrical viewpoint. In this paper, we prove the local well-posedness of the problem in spirit of the arguments developed recently by Gu-Wang [14]. Our method is based on the use of Alinhac good unknowns, the suitable approximations both nonlinear and linear, and delicate energy estimates.

A time domain finite element approach based on time series estimation

This paper details the use of time series for solving finite element equations. It was aimed at eliminating the discretization of time intervals to reduce calculation time and to obtain accurate results using time-dependent variables. For this purpose, the matrix form of the time series finite element formulation for both continuous and discontinuous time functions were extracted using Taylor and Fourier series for estimating time-dependent functions. Based on the extracted formulation, for solving elastodynamic problems, a computer code and algorithm was introduced. The efficiency and accuracy of the formulation was evaluated by solving different analytical examples. To reveal the power of time series for modeling real problems three practical examples such as settlement, flood protector structures and temporary cofferdams were modeled using time series approach and the results of analyses were presented. The results showed that having removed time discretization procedure, the time series based finite element method can be efficiently applied to decrease the calculation time and computational efforts. In addition, time-continuous responses resulting from this method, help reduce the calculation steps which is vital in many engineering problems.

Cell‐based maximum entropy approximants for three‐dimensional domains: Application in large strain elastodynamics using the meshless total Lagrangian explicit dynamics method

SummaryWe present the cell‐based maximum entropy (CME) approximants in E3 space by constructing the smooth approximation distance function to polyhedral surfaces. CME is a meshfree approximation method combining the properties of the maximum entropy approximants and the compact support of element‐based interpolants. The method is evaluated in problems of large strain elastodynamics for three‐dimensional (3D) continua using the well‐established meshless total Lagrangian explicit dynamics method. The accuracy and efficiency of the method is assessed in several numerical examples in terms of computational time, accuracy in boundary conditions imposition, and strain energy density error. Due to the smoothness of CME basis functions, the numerical stability in explicit time integration is preserved for large time step. The challenging task of essential boundary condition (EBC) imposition in noninterpolating meshless methods (eg, moving least squares) is eliminated in CME due to the weak Kronecker‐delta property. The EBCs are imposed directly, similar to the finite element method. CME is proven a valuable alternative to other meshless and element‐based methods for large‐scale elastodynamics in 3D. A naive implementation of the CME approximants in E3 is available to download at https://www.mountris.org/software/mlab/cme.

Numerical recipes for elastodynamic virtual element methods with explicit time integration

SummaryWe present a general framework to solve elastodynamic problems by means of the virtual element method (VEM) with explicit time integration. In particular, the VEM is extended to analyze nearly incompressible solids using the B‐bar method. We show that, to establish a B‐bar formulation in the VEM setting, one simply needs to modify the stability term to stabilize only the deviatoric part of the stiffness matrix, which requires no additional computational effort. Convergence of the numerical solution is addressed in relation to stability, mass lumping scheme, element size, and distortion of arbitrary elements, either convex or nonconvex. For the estimation of the critical time step, two approaches are presented, ie, the maximum eigenvalue of a system of mass and stiffness matrices and an effective element length. Computational results demonstrate that small edges on convex polygonal elements do not significantly affect the critical time step, whereas convergence of the VEM solution is observed regardless of the stability term and the element shape in both two and three dimensions. This extensive investigation provides numerical recipes for elastodynamic VEMs with explicit time integration and related problems.