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The paper concerns numerical study of non-Fourier bio heat transfer model in multi-layered media. Specifically, we employ the Maxwell–Cattaneo equation on the physical media with discontinuous coefficients. A fitted finite element method is proposed and analyzed for a hyperbolic heat conduction model with discontinuous coefficients. Typical semidiscrete and fully discrete schemes are presented for a fitted finite element discretization with straight interface triangles. The fully discrete space–time finite element discretizations are based on second order in time Newmark scheme. Optimal a priori error estimates for both semidiscrete and fully discrete schemes are proved in L ∞ ( L 2 ) norm. Numerical experiments are reported for several test cases to confirm our theoretical convergence rate. Finite element algorithm presented here can be used to solve a wide variety of hyperbolic heat conduction models for non-homogeneous inner structures.

Purpose The purpose of this paper is to propose a local parallel finite element algorithm based on fully overlapping domain decomposition technique to solve the incompressible magnetohydrodynamic equations. Design/methodology/approach The algorithm uses a lower-order element pair to compute an initial approximation by the Oseen-type iteration and uses a higher-order element pair to solve a linear system in each processor. Findings Besides, the convergence analysis of local parallel finite element algorithm is given. Finally, numerical experiments are presented to verify the efficiency of the proposed algorithm. Originality/value Compared with the numerical solution of the common two-step method, this method is easy to realize and can produce a more accurate solution. And, this approach is executed in parallel, so it saves a lot of computational time.

Based on full domain partition, parallel finite element algorithm based on three linearizations is designed to solve the stationary incompressible magnetohydrodynamics (MHD). In every algorithm, each processor is in charge of a subproblem which is linearized by three options: Stokes-type, Newton’s and Picard’s iterations. Each subproblem is defined in the whole domain with vast majority of the degrees of freedom associated with the particular subdomain. The error estimates of these proposed algorithms with respect to iterative step and small mesh sizes are analyzed. Finally, some numerical experiments are provided to show the validity and high efficiency of our algorithms.

In this paper, finite element method with high-order approximation for time fractional derivative is considered and discussed to find the numerical solution of time fractional convection-diffusion equation. Some lemmas are introduced and proved, further the stability and error estimates are discussed and analyzed, respectively. The convergence result O(hr+1 + τ3-α) can be derived, which illustrates that time convergence rate is higher than the order (2-α) derived by L1-approximation. Finally, to validate our theoretical results, some computing data are provided.

An efficient two-level finite element algorithm for solving the natural convection equations is developed and studied in this paper. By solving one small nonlinear system on a coarse mesh H and two large linearized problems on a fine mesh h = O ( H 7 − e 2 ) with different loads, we can obtain an approximation solution ( u h , p h , T h ) with the convergence rate of same order as the usual finite element solution, which involves one large nonlinear natural convection system on the same fine mesh h. Furthermore, compared with the results of Si's algorithm in 2011, the given algorithm costs less computed time to get almost the same precision.

In this paper, we prove the convergence of an adaptive finite element method for optimal control problems on $$L^{2}$$L2 errors by keeping the meshes sufficiently mildly. In order to keep the meshes sufficiently mildly we need increasing the number of elements that are refined, moreover, we find that it will not compromise the quasi-optimality of the AFEM. In other words, we prove the quasi-optimality of the adaptive finite element algorithm in the present paper. In the end, we conclude this paper with some conclusions and future works.

This chapter presents the basic principles to be taken into account for the application of the stress–strain analysis methodology. The algorithms most commonly used in stress–strain analysis are the boundary elements, finite elements and finite difference elements algorithms. The finite element algorithm shares its name with the stress–strain calculation methodology, since when this methodology began being developed in the 1960s, there was only one algorithm to solve the equations, and therefore, the mathematical algorithm used kept this name. The finite element algorithm has another important disadvantage, namely, the way in which the models with intense yielding are solved. The distinct element algorithm is an evolution of the finite difference algorithm, which has been created to solve stress–strain calculation problems in a discontinuous medium with elasto-plastic behavior. Stress–strain calculation software programs are very powerful tools to accurately analyze the redistribution process of the stresses in the ground when an underground excavation is constructed.

In this work, we are concerned with the local and parallel finite element algorithm based on the Oseen-type iteration for solving the stationary incompressible magnetohydrodynamics. Under the uniqueness condition, the error estimates with respect to iterative step m and small mesh sizes H and $$h\ll H$$hźH of the proposed method are derived. Finally, some numerical experiments are provided to show the high efficiency of our algorithm.

The new second-order two-scale (SOTS) finite element algorithm is developed for the dynamic thermo-mechanical coupling problems in axisymmetric and spherical symmetric structures made of composite materials. The axisymmetric structure considered is periodic in both radial and axial directions and homogeneous in circumferential direction. The spherical symmetric structure is only periodic in radial direction. The dynamic thermo-mechanical coupling model is presented and the equivalent compact form is derived. Then, the cell problems, effective material coefficients and the homogenized thermo-mechanical coupling problem are obtained successively by the second-order asymptotic expansion of the temperature increment and displacement. The homogenized material obtained is manifested with the anisotropic property in the circumferential direction. The explicit expressions of the homogenized coefficients in the plane axisymmetric and spherical symmetric cases are given and both the derivation of the analytical solutions of the cell functions and the quasi-static thermoelasticity problems are discussed. Based on the SOTS method, the corresponding finite-element procedure is presented and the unconditionally stable implicit algorithm is established. Some numerical examples are solved and the mutual interaction between the temperature and displacement field is studied under the condition of structural vibration. The computational results demonstrate that the second-order asymptotic analysis finite-element algorithm is feasible and effective in simulating and predicting the dynamic thermo-mechanical behaviors of the composite materials with small periodic configurations in axisymmetric and spherical symmetric structures. This may provide a vital computational tool for analyzing composite material internal temperature distribution and structural deformation induced by the dynamic thermo-mechanical coupling response under strong aerothermodynamic environment. Second-order two-scale (SOTS) expansions for dynamic thermo-mechanical coupling problems in symmetric structure are obtained.SOTS finite element algorithm is proposed and unconditional stable implicit scheme is established.Anisotropic material is obtained by homogenization with enhanced strength and minor thermal expansion in circumferential direction.The mutual interaction and simultaneous vibration of the temperature and displacement field are simulated.A new computational tool is presented for analyzing thermo-mechanical response of composites under strong aerothermodynamic environment.

A new second-order two-scale (SOTS) analysis finite element algorithm is developed for the axisymmetric and spherical symmetric elastic problems with small periodic configurations. The axisymmetric structure considered is periodic in both radial and axial directions and homogeneous in circumferential direction, and the spherical symmetric structure is only periodic in radial direction and homogeneous in other two directions. The SOTS asymptotic expansions for the space problem, plane axisymmetric problem, and spherical symmetric problem are presented, and the main feature is that the anisotropic material is obtained by the homogenization. The analytical expressions of the cell functions and homogenized solutions for plane axisymmetric and spherical symmetric problems are obtained, and the error estimations of the expansions are established. The second-order asymptotic analysis finite-element algorithm is presented and the numerical examples are solved including the hollow cylinder, rotating disk and hollow sphere composed of periodic composite materials. The computational results demonstrate the effectiveness and accuracy of the SOTS asymptotic analysis algorithm, and the converging behavior of the asymptotic analysis algorithm agrees well with the theoretical prediction. It is also indicated that the stress distributions can be correctly computed only by adding the second-order correctors.

An adaptive finite element algorithm for problems in elastoplasticity with hardening is proven to be of optimal convergence with respect to the notion of approximation classes. The results rely on the equivalence of the errors of the stresses and energies resulting from Jensen's inequality. Numerical experiments study the influence of the hardening and bulk parameters to the convergence behavior of the AFEM algorithm. This is the first optimal adaptive FEM for a variational inequality.

In this paper, an adaptive fully discrete finite element algorithm of semilinear parabolic optimal control problems is investigated. On deriving a posteriori error estimates, we construct an adaptive finite element algorithm for parabolic optimal control problems. Some numerical examples are presented. It confirms that our adaptive algorithm is efficient and our theoretical results are correct.

Inspired by recent analyses of the finite element heterogeneous multiscale method and the reduced basis technique for nonlinear problems, we present a simple and concise finite element algorithm for the reliable and efficient resolution of elliptic or parabolic multiscale problems of nonmonotone type. Solutions of appropriate cell problems on sampling domains are selected by a greedy algorithm in an offline stage and assembled in a reduced basis (RB). This RB is then used in an online stage to solve two-scale problems at a computational cost comparable to the single-scale case. Both the offline and the online cost are independent of the smallest scale in the physical problem. The performance and accuracy of the algorithm are illustrated on 2D and 3D stationary and evolutionary nonlinear multiscale problems.

A combination method of the Newton iteration and the two-level stabilized finite element algorithm based on local Gauss integration is constructed for solving numerically the steady Smagorinsky model. This algorithm involves solving one small, nonlinear coarse mesh with mesh size H and two linear problems on the fine mesh with mesh size h . Based on the stabilized method and the Newton two-level technique, the computation will be more effective and convenient and the scaling between H and h becomes h = O ( H 4 ) , which greatly complements the results of Borggaard et al. (2008) [2] . Moreover, the stability and convergence of the two-level Newton iterative solution are analyzed. Finally, some numerical tests are made to demonstrate the effectiveness of the given method.

In this paper, we propose an adaptive finite element algorithm for the numerical solution of a class of nonlocal models which correspond to nonlocal diffusion equations and linear scalar peridynamic models with certain nonintegrable kernel functions. The convergence of the adaptive finite element algorithm is rigorously derived with the help of several basic ingredients, such as the upper bound of the estimator, the estimator reduction, and the orthogonality property. We also consider how the results are affected by the horizon parameter $\delta$ which characterizes the range of nonlocality. Numerical experiments are performed to verify our theoretical findings.

This paper presents an optimal nonconforming adaptive finite element algorithm and proves its quasi-optimal complexity for the Stokes equations with respect to natural approximation classes. The proof does not explicitly involve the pressure variable and follows from a novel discrete Helmholtz decomposition of deviatoric functions.

The nonlocal continuum dielectric model is an important extension of the classical Poisson dielectric model, but it is very expensive to be solved in general. In this paper, we prove that the solution of one commonly used nonlocal continuum dielectric model of water can be split as a sum of two functions, and these two functions are simply the solutions of one Poisson equation and one Poisson-like equation. With this new solution splitting formula, we develop a fast finite element algorithm and a program package in Python based on the DOLFIN program library such that a nonlocal dielectric model can be solved numerically in an amount of computation that merely doubles that of solving a classic Poisson dielectric model. Using the new solution splitting formula, we also derive the analytical solutions of two nonlocal model problems. We then solve these two nonlocal model problems numerically by our program package and validate the numerical solutions through a comparison with the analytical solutions. Finally, our study of free energy calculation by a nonlocal Born ion model demonstrates that the nonlocal dielectric model is a much better predictor of the solvation free energy of ions than the local Poisson dielectric model.

Based on the full domain partition, a parallel finite element algorithm for the stationary Stokes equations is proposed and analyzed. In this algorithm, each subproblem is defined in the entire domain. Majority of the degrees of freedom are associated with the relevant subdomain. Therefore, it can be solved in parallel with other subproblems using an existing sequential solver without extensive recoding. This allows the algorithm to be implemented easily with low communication costs. Numerical results are given showing the high efficiency of the parallel algorithm.

A moving mesh finite element method is proposed for the adaptive solution of second- and fourth-order moving boundary problems which exhibit scale invariance. The equations for the mesh movement are based upon the local application of a scale-invariant conservation principle incorporating a monitor function and have been successfully applied to problems in both one and two space dimensions. Examples are provided to show the performance of the proposed algorithm using a monitor function based upon arc-length.

This paper is focused on the reanalysis of structures with added degrees of freedom. By suitably partitioning the extended stiffness matrix, we present a preconditioned conjugate gradient algorithm to analyze the modified structure. The algorithm preserves the ease of implementation and improves significantly the quality of results. In particular, the algorithm can adaptively monitor the accuracy of approximate solutions, it is applicable to general finite element systems. The computational cost is significantly reduced. Numerical examples are given to show the effectiveness of the presented reanalysis algorithm.

The two-scale asymptotic (TSA) expressions of the increment of temperature and the displacement for the structure of composite materials with small periodic configuration under coupled thermoelasticity condition are derived formally in this paper, especially, the two-scale coupled relation between the increment of temperature and the displacements are set up. Then the approximate solutions and their error estimations are presented, and the multi-scale finite element algorithm corresponding to TSA is described. Finally, simple numerical results evaluated by multi-scale FE computation are shown. They demonstrate that the basic configuration and the increment of temperature strongly influence upon local strains and local stresses inside basic cell. Copyright © 2004 John Wiley & Sons, Ltd.

We present and investigate a model for the thermomechanical behaviour of a viscoelastic beam with an attached tip body. The model consists of a coupled energy–elastic system of equations for the beam together with an equation for the temperature of the tip body. We establish existence and uniqueness of a weak solution to the system. We also propose a finite element algorithm for solutions to the system and present numerical simulations based on the algorithm.

A formulation is presented herein for implementing a cohesive zone model to a nonlinear finite element algorithm. The cohesive zone model may be of arbitrary type so long as it can be constructed in an incremental form in time. Problems that can then be solved using this algorithm include a broad array of material types, including: elastic, elastoplastic, viscoplastic, and viscoelastic. Multiple cracks may be modeled by this methodology, and any and all can be evolving simultaneously, with crack interaction included explicitly in the formulation. Example problems are included to demonstrate the efficacy of the algorithm.

We establish Crouzeix-Raviart element adaptive algorithm based on Rayleigh quotient iteration and give its a priori/a posteriori error estimates. Our algorithm is performed under the package of Chen, and satisfactory numerical results are obtained.

A hybrid monotonous finite element algorithm is developed in the present paper, based on a second-order-accurate finite element scheme and a first-order-accurate monotonous one derived from the former by a unilateral lumping procedure in one dimensional case. The switch functions for the two dimensional Euler equation system are constructed locally, based on the gradient of the flow field, with special consideration on the information from neighboring elements. Examples show that the ew scheme can eliminate oscillations near strong shocks obviously.

A proposed destructive technique to determine residual stresses in thick-walled axisymmetric pressure vessels employs incremental strain gauge data in a finite element algorithm to construct the initial residual stress distribution. The data acquisition component of the proposed technique has similarities with Sachs' bore out method, enabling independently obtained data to be employed by the finite element algorithm. A comparison between the two methods based on the same strain data enables the accuracy of the finite element algorithm to be assessed. Results to date suggest that the new algorithm determines residual stress distributions to an accuracy comparable with Sachs' bore out method. An advantage that the proposed technique has over Sachs' bore out method is that it can be applied to axisymmetric thick-walled pressure vessels of geometries more complex than plain cylinders.

A finite element algorithm for consideration of contact constraints is presented. It is characterized by introducing the geometric constraints, resulting from contact conditions, directly into the algebraic system of equations for the incremental displacements of an incremental iterative solution procedure. The usefulness of the proposed algorithm for efficient solutions of contact problems involving large displacements and large strains is demonstrated in the numerical investigation.

A concept for the analysis of viscoplastic structures is introduced that allows to develop viscoplastic material models for steel based on “established” elastic-plastic constitutive equations. The attainment of an exact limiting value for quasistatic processes permits utilizing an efficient strategy to determine the material functions. Structural analyses are conducted with an iterative and incremental finite element algorithm. The efficiency of the proposed concept is assessed by comparison with experimental and numerical results taken from the literature.

We present the results of some numerical experiments which were carried out in order to investigate the general characteristics of the algorithm described in Part I of this paper.

We study the surface tension driven leveling of surface irregularities in viscoelastic films deposited on a flat surface. Analytical results are presented for the generalized Maxwell model in the limit of infinitesimal surface disturbances. The results show that elasticity retards the leveling process. In some cases, the retardation effect is dramatic. Keunings' finite element algorithm for solving viscoelastic moving boundary problems is used to analyze the case of finite-amplitude surface disturbances. Results for an Oldroyd B fluid display significant elastic effects which are consistent with the asymptotic predictions.

A series of implicit finite element algorithms for the geometrically nonlinear structural dynamics problem are proposed. The proposed algorithms require only the solution of a linear system at each time step. Thus, they are computationally efficient. In addition, the algorithms are discrete conservation laws. The conservative nature of the proposed schemes has a positive effect in providing a stable approximation. The stability of the algorithms is analyzed using energy methods. One of the proposed computational methods is shown to be unconditionally stable.

A new method of performing the shakedown analysis of structures subjected to variable repeated loads and temperature fields is presented. The method employs the standard incremental elastic-plastic finite element algorithm and is based upon a cyclic repetition of piecewise-proportional paths that contain all the vertices of the load-temperature variation domain. A numerical example illustrates the approach.

A least square based finite element algorithm is developed for some elasto-static problems. In the formulation both stresses and displacements appear as simultaneous variables. In two dimensional (plane) analysis, parabolie isoparametric elements are used. Considerable improvement of performance is obtained with a numerical integration based on 2 × 2 Gauss point distribution over more accurate integration schemes. Reasons for this are presented. The formulation is extended in the section ‘General least square formulation’ to beams and plates with a similar success of ‘reduced’ integration.

The nonlinear diffusion equation as applied to two-dimensional time-dependent magnetic fields is solved with a finite element algorithm. This algorithm permits the analysis of problems possessing complex geometries, induced eddy currents, permanent magnets, and nonperiodic excitation currents. The numerical procedure utilizes implicit time stepping with an iterative scheme to solve the resulting set of equations. Two examples of applications of this program are presented.

A consideration of alternative phenomena has been undertaken as a next logical step in the continuing development of a finite element model of wave-defect interactions [1,2]. Preliminary investigation indicates that obtaining the requisite empirical data offers serious challenges and the current study has therefore focused on the necessity of introducing explicit alternative components into the existing algorithm. A brief review of attenuati ve processes is provided as a means of considering numerical modeling approaches. An axi-symmetric finite element algorithm is subsequently introduced and evaluated with experimentally obtained data. The algorithm’s three-dimensional nature facilitates such an evaluation and represents a significant step in the development of a quantitatively accurate model of wave-defect interactions. Conclusions are then made about the types of materials for which this simple elastic model is adequate.