Elastodynamic Problems Research Articles

The time-dependent Koiter model and its numerical computation

• Existence and uniqueness of the time-dependent Koiter model. • Newmark scheme for the time-dependent Koiter model. • Analyses of existence, uniqueness, stability, convergence and priori error estimate for elastodynamic problems. The theory of elastic shells is one of the most important branches of the theory of elasticity. Among all the shell models, a classical and widely recognized model is the Koiter model. In this paper, we discuss the time-dependent Koiter model, which has not been addressed numerically. We show that the solution of this model exists and is unique. We semi-discretize the space variable and fully discretize the problem using the time discretization by the Newmark scheme. The corresponding analyses of existence, uniqueness, stability, convergence and a priori error estimate are given. Finally, we provide numerical experiments with a portion of spherical shell and a portion of cylindrical shell to demonstrate the efficiency of the model.

A stiffly accurate integrator for elastodynamic problems

We present a new integration algorithm for the accurate and efficient solution of stiff elastodynamic problems governed by the second-order ordinary differential equations of structural mechanics. Current methods have the shortcoming that their performance is highly dependent on the numerical stiffness of the underlying system that often leads to unrealistic behavior or a significant loss of efficiency. To overcome these limitations, we present a new integration method which is based on a mathematical reformulation of the underlying differential equations, an exponential treatment of the full nonlinear forcing operator as opposed to more standard partially implicit or exponential approaches, and the utilization of the concept of stiff accuracy which ensures that the efficiency of the simulations is significantly less sensitive to increased stiffness. As a consequence, we are able to tremendously accelerate the simulation of stiff systems compared to established integrators and significantly increase the overall accuracy. The advantageous behavior of this approach is demonstrated on a broad spectrum of complex examples like deformable bodies, textiles, bristles, and human hair. Our easily parallelizable integrator enables more complex and realistic models to be explored in visual computing without compromising efficiency.

The meshless finite point method for transient elastodynamic problems

In this paper, the application of the meshless finite point method (FPM) to solve elastodynamic problems through an explicit velocity–Verlet time integration method is investigated. Strong form-based methods, such as the FPM, are generally less stable and accurate in terms of satisfaction of Neumann boundary conditions than weak form-based methods. This is due to the fact that in such types of methods, Neumann boundary conditions must be imposed by a series of equations which are different from the governing equations in the problem domain. In this paper, keeping all the advantages of FPM in terms of simplicity and efficiency, a new simple strategy for proper satisfaction of Neumann boundary conditions in time for elastodynamic problems is investigated. The method is described in detail, and several numerical examples are presented. Moreover, the accuracy of the method with reference to the solution of some 3D problems is discussed.

The solution of elastostatic and dynamic problems using the boundary element method based on spherical Hankel element framework

SummaryIn this paper, new spherical Hankel shape functions are used to reformulate boundary element method for 2‐dimensional elastostatic and elastodynamic problems. To this end, the dual reciprocity boundary element method is reconsidered by using new spherical Hankel shape functions to approximate the state variables (displacements and tractions) of Navier's differential equation. Using enrichment of a class of radial basis functions (RBFs), called spherical Hankel RBFs hereafter, the interpolation functions of a Hankel boundary element framework has been derived. For this purpose, polynomial terms are added to the functional expansion that only uses spherical Hankel RBF in the approximation. In addition to polynomial function fields, the participation of spherical Bessel function fields has also increased robustness and efficiency in the interpolation. It is very interesting that there is no Runge phenomenon in equispaced Hankel macroelements, unlike equispaced classic Lagrange ones. Several numerical examples are provided to demonstrate the effectiveness, robustness and accuracy of the proposed Hankel shape functions and in comparison with the classic Lagrange ones, they show much more accurate and stable results.

A formulation for elastodynamic scattering based on boundary algebraic equations

The solution of the elastodynamic problem arises in many scientific fields such as the wave propagation in the ground, non-destructive testing, the vibration design of buildings or vibroacoustics in general. We present here an integral formulation based on the boundary algebraic equations. It leads to a boundary numerical method and has the important advantage that no contour (2D) or surface (3D) integrals need to be computed. This is very helpful in order to use combined field integral equations (numerical damping of fictitious eigenfrequencies) without the problems caused by the evaluation of hypersingular integrals. The key aspects are: (i) the approach deals with discrete equations from the very beginning; (ii) discrete (instead of continuous) tensor Green’s functions are considered (the methodology to evaluate them is shown); (iii) the boundary must be described by means of a regular square grid. In order to overcome the drawback of this third condition the boundary integral is coupled, if needed, with a thin layer of finite elements. This improves the description of curved geometries and reduces numerical errors. The method is applied to the scattering of waves by objects and holes in an unbounded elastic medium and to the solution of interior elastic problems

Complementary energy principle for elastodynamics: Free of volumetric locking

This paper proposes a complementary energy principle for elastodynamic problems. In doing so, the stresses and the inertial forces, that satisfy the equilibrium equations a priori, are chosen as basic variables and then, the complementary variational formulation is established by weakly enforcing the kinematic constraints. The eventual formulation is posed to be an integral equation with respect to the time dimension and can be easily applied to free and forced vibration analysis. In addition, the volumetric locking-free property inherent in the complementary energy principle is addressed. Numerical study on a rectangular plate is performed to validate the proposed complementary energy principle as well as to verify the featured locking-free property. Moreover, extensions of the dynamic complementary energy principle to tackle visco-elastodynamics and Stokes flows are also presented.

Lower Bound of Vibration Modes Using the Node-Based Smoothed Finite Element Method (NS-FEM)

The smoothed finite element method (S-FEM) has been recently developed as an effective solver for solid mechanics problems. This paper represents an effective approach to compute the lower bounds of vibration modes or eigenvalues of elasto-dynamic problems, by making use of the important softening effects of node-based S-FEM (NS-FEM). We first use NS-FEM, FEM and the analytic approach to compute the eigenvalues of transverse free vibration in strings and membranes. It is found that eigenvalues by NS-FEM are always smaller than those by FEM and the analytic method. However, NS-FEM produces spurious unphysical modes because of overly soft behavior. A technique is then proposed to remove them by analyzing their vibration shapes (eigenvectors). It is observed that spurious modes with excessively large wave numbers, which are unrelated to the physical deflection shapes but related to the discretization density, therefore can be easily removed. The final results of NS-FEM become the lower bound of eigenvalues and the accuracy can be improved via mesh refinement. And NS-FEM solutions (softer) are more reliable, because the large wave number component can be used as an indicator, which is available in FEM (stiffer), on the quality of the numerical solutions. The proposed NS-FEM procedure offers a viable and practical computational means to effectively compute the lower bounds of eigenvalues for solid mechanics problems.

Fast iterative boundary element methods for high-frequency scattering problems in 3D elastodynamics

The fast multipole method is an efficient technique to accelerate the solution of large scale 3D scattering problems with boundary integral equations. However, the fast multipole accelerated boundary element method (FM-BEM) is intrinsically based on an iterative solver. It has been shown that the number of iterations can significantly hinder the overall efficiency of the FM-BEM. The derivation of robust preconditioners for FM-BEM is now inevitable to increase the size of the problems that can be considered. The main constraint in the context of the FM-BEM is that the complete system is not assembled to reduce computational times and memory requirements. Analytic preconditioners offer a very interesting strategy by improving the spectral properties of the boundary integral equations ahead from the discretization. The main contribution of this paper is to combine an approximate adjoint Dirichlet to Neumann (DtN) map as an analytic preconditioner with a FM-BEM solver to treat Dirichlet exterior scattering problems in 3D elasticity. The approximations of the adjoint DtN map are derived using tools proposed in [40]. The resulting boundary integral equations are preconditioned Combined Field Integral Equations (CFIEs). We provide various numerical illustrations of the efficiency of the method for different smooth and non-smooth geometries. In particular, the number of iterations is shown to be completely independent of the number of degrees of freedom and of the frequency for convex obstacles.

A quadratic time finite element method for nonlinear elastodynamics within the context of hybrid finite elements

Standard displacement-based finite elements are known to display overstiff behavior known as locking phenomenon for a wide variety of problems. To overcome this problem, hybrid elements have been proposed in the literature. They are based on a two-field variational principle, and involve an interpolation for the stress field that is chosen independently of that of the displacement field. Due to this independent choice of the stress interpolation, hybrid elements are less susceptible to the locking phenomenon, and thus provide better coarse mesh accuracy as compared to the displacement-based formulation. Hybrid elements are versatile elements with which one can model shell structures, almost incompressible materials as well as ‘chunky’ geometries. In this work, we use hybrid elements for the space discretization along with a higher-order momentum conserving variant of the time finite element method. The time finite element method provides a general variational framework for developing arbitrary-order time-stepping strategies for transient problems. It has been shown in the literature recently that for discrete chaotic systems the quadratic time finite element scheme is computationally more efficient and robust when compared to the linear time finite element strategy. In this work, we present a modified form of the quadratic time finite element method for the solution of nonlinear elastodynamics problems that conserves linear and angular momenta exactly, and energy in an approximate sense in the fully discrete setting. We present numerous examples to demonstrate the efficacy of hybrid elements over conventional elements within the context of the modified quadratic time finite element method. From our examples, we observe that for chaotic systems, the proposed strategy is significantly more computationally efficient compared to the linear transient strategy; however in other problems although the performance is good, it does not offer any significant computational advantage.

Robust Semi-Discrete and Fully Discrete Hybrid Stress Finite Element Methods for Elastodynamic Problems

AbstractThis paper analyzes semi-discrete and fully discrete hybrid stress quadrilateral finite element methods for 2-dimensional linear elastodynamic problems. The methods use a 4 node hybrid stress quadrilateral element in the space discretization. In the fully discrete scheme, an implicit second-order scheme is adopted in the time discretization. We derive optimal a priori error estimates for the two schemes and an unconditional stability result for the fully discrete scheme. Numerical experiments confirm the theoretical results.

Consideration related to integral equation formlations for transmission problems of elastodynamics in 2D

Consideration related to integral equation formlations for transmission problems of elastodynamics in 2D

A multisymplectic integrator for elastodynamic frictionless impact problems

We present a structure preserving numerical algorithm for the collision of elastic bodies. Our integrator is derived from a discrete version of the field-theoretic (multisymplectic) variational description of nonsmooth Lagrangian continuum mechanics, combined with generalized Lagrange multipliers to handle inequality constraints. We test the resulting explicit integrator for the longitudinal impact of two elastic linear bar models, and for the collision of a nonlinear geometrically exact beam model with a rigid plane. Numerical simulations for various physical parameters are presented to illustrate the behavior and performance of our approach.

A coupled meshless finite point/Peridynamic method for 2D dynamic fracture analysis

In this paper, a new coupled meshless method for 2D transient elastodynamic problems involving dynamic crack propagation is developed. The method is based on an efficient coupling between the finite point method (FPM) and a discretized form of Peridynamics. The solution domain is partitioned in three parts: one discretized by Peridynamics, one by FPM, and a transition part discretized by both methods where the switching between the two approaches is performed. The coupling adopts a local/nonlocal framework that benefits from the full advantages of both methods. The parts of domain where cracks either exist or are likely to propagate are described by Peridynamics; the remaining part of the domain is described by FPM that requires less computational effort. The capabilities of the proposed approach are demonstrated by means of the solution of dynamic problems including dynamic fracture as well as a ghost force test.

C1‐continuous space‐time discretization based on Hamilton's law of varying action

We develop a class of C1‐continuous time integration methods that are applicable to conservative problems in elastodynamics. These methods are based on Hamilton's law of varying action. From the action of the continuous system we derive a spatially and temporally weak form of the governing equilibrium equations. This expression is first discretized in space, considering standard finite elements. The resulting system is then discretized in time, approximating the displacement by piecewise cubic Hermite shape functions. Within the time domain we thus achieve C1‐continuity for the displacement field and C0‐continuity for the velocity field. From the discrete virtual action we finally construct a class of one‐step schemes. These methods are examined both analytically and numerically. Here, we study both linear and nonlinear systems as well as inherently continuous and discrete structures. In the numerical examples we focus on one‐dimensional applications. The provided theory, however, is general and valid also for problems in 2D or 3D. We show that the most favorable candidate — denoted as p2‐scheme — converges with order four. Thus, especially if high accuracy of the numerical solution is required, this scheme can be more efficient than methods of lower order. It further exhibits, for linear simple problems, properties similar to variational integrators, such as symplecticity. While it remains to be investigated whether symplecticity holds for arbitrary systems, all our numerical results show an excellent long‐term energy behavior.

Numerical Laplace inversion in problems of elastodynamics: Comparison of four algorithms

The objective of this work is to find a suitable algorithm for numerical Laplace inversion which could be used for effective and precise solution of elastodynamic problems. For this purpose, the capabilities of four algorithms are studied using three transforms resulted from analytical solutions of longitudinal waves in a thin rod, flexural waves in a thin beam and plane waves in a strip. In particular, the Gaver–Stehfest algorithm, the Gaver–Wynn’s rho algorithm, the Fixed-Talbot algorithm and the FFT algorithm combined with Wynn’s epsilon accelerator are tested. The codes written in Maple 16 employing multi-precision computations are presented for each method. Given the results obtained, the last mentioned algorithm proves to be the best. It is most efficient and it gives results of reasonable accuracy nearly for all tested times ranging from 3×10−7s to 3 × 103 s.

A spectral dynamic stiffness method for free vibration analysis of plane elastodynamic problems

A highly efficient and accurate analytical spectral dynamic stiffness (SDS) method for modal analysis of plane elastodynamic problems based on both plane stress and plane strain assumptions is presented in this paper. First, the general solution satisfying the governing differential equation exactly is derived by applying two types of one-dimensional modified Fourier series. Then the SDS matrix for an element is formulated symbolically using the general solution. The SDS matrices are assembled directly in a similar way to that of the finite element method, demonstrating the method's capability to model complex structures. Any arbitrary boundary conditions are represented accurately in the form of the modified Fourier series. The Wittrick-Williams algorithm is then used as the solution technique where the mode count problem (J0) of a fully-clamped element is resolved. The proposed method gives highly accurate solutions with remarkable computational efficiency, covering low, medium and high frequency ranges. The method is applied to both plane stress and plane strain problems with simple as well as complex geometries. All results from the theory in this paper are accurate up to the last figures quoted to serve as benchmarks.

Numerical evaluation of integrals involving the product of two Bessel functions and a rational fraction arising in some elastodynamic problems

This paper presents numerical techniques for evaluating integrals of the form ∫0∞Jβ(kρ)Jγ(kR)kα(k−s)dk. These integrals arise during the application of the Hankel transform to pass the displacements of a layered soil profile from the wave number domain to the spatial domain in three-dimensional problems of elastodynamics. The objective here is to obtain solutions with an adequate accuracy from the engineering point of view to the integrals that arise in a first order formulation of a wave propagation model widely used for layered soils.

Scattering problems in elastodynamics

In electromagnetism, acoustics, and quantum mechanics, scattering problems can routinely be solved numerically by virtue of perfectly matched layers (PMLs) at simulation domain boundaries. Unfortunately, the same has not been possible for general elastodynamic wave problems in continuum mechanics. In this paper, we introduce a corresponding scattered-field formulation for the Navier equation. We derive PMLs based on complex-valued coordinate transformations leading to Cosserat elasticity-tensor distributions not obeying the minor symmetries. These layers are shown to work in two dimensions, for all polarizations, and all directions. By adaptative choice of the decay length, the deep subwavelength PMLs can be used all the way to the quasi-static regime. As demanding examples, we study the effectiveness of cylindrical elastodynamic cloaks of the Cosserat type and approximations thereof.

A fully smoothed XFEM for analysis of axisymmetric problems with weak discontinuities

SummaryIn this paper, we propose a fully smoothed extended finite element method for axisymmetric problems with weak discontinuities. The salient feature of the proposed approach is that all the terms in the stiffness and mass matrices can be computed by smoothing technique. This is accomplished by combining the Gaussian divergence theorem with the evaluation of indefinite integral based on smoothing technique, which is used to transform the domain integral into boundary integral. The proposed technique completely eliminates the need for isoparametric mapping and the computing of Jacobian matrix even for the mass matrix. When employed over the enriched elements, the proposed technique does not require sub‐triangulation for the purpose of numerical integration. The accuracy and convergence properties of the proposed technique are demonstrated with a few problems in elastostatics and elastodynamics with weak discontinuities. It can be seen that the proposed technique yields stable and accurate solutions and is less sensitive to mesh distortion. Copyright © 2016 John Wiley & Sons, Ltd.

Evaluation of the T-stress for cracks in functionally graded materials by the FEM

Using the quarter-point crack tip element, a simple formula is derived which provides a quick and expedient means to evaluate the elastic T-stress in the finite element method (FEM) fracture mechanics analysis of functionally graded materials (FGMs). The formula is derived by comparing the variation of displacements in the quarter-point element with the corresponding asymptotic expression for the displacement field in the vicinity of a crack-tip. The asymptotic fields in the crack-tip vicinity in a FGM with continuously varying material properties are the same as in a homogeneous one. The simplicity of the use of this formula for T-stress evaluation in FEM fracture mechanics analysis is demonstrated by its application to some transient elastodynamic and thermoelastic problems.