A partition of unity FEM for time‐dependent diffusion problems using multiple enrichment functions

An enriched partition of unity FEM is developed to solve time‐dependent diffusion problems. In the present formulation, multiple exponential functions describing the spatial and temporal diffusion decay are embedded in the finite element approximation space. The resulting enrichment is in the form of a local asymptotic expansion. Unlike previous works in this area where the enrichment must be updated at each time step, here, the temporal decay in the solution is embedded in the asymptotic expansion. Thus, the system matrix that is evaluated at the first time step may be decomposed and retained for every next time step by just updating the right‐hand side of the linear system of equations. The advantage is a significant saving in the computational effort where, previously, the linear system must be reevaluated and resolved at every time step. In comparison with the traditional finite element analysis with p‐version refinements, the present approach is much simpler, more efficient, and yields more accurate solutions for a prescribed number of DoFs. Numerical results are presented for a transient diffusion equation with known analytical solution. The performance of the method is analyzed on two applications: the transient heat equation with a single source and the transient heat equation with multiple sources. The aim of such a method compared with the classical FEM is to solve time‐dependent diffusion applications efficiently and with an appropriate level of accuracy. Copyright © 2012 John Wiley & Sons, Ltd.     

Transient shell response by numerical time integration

In using the finite element method to compute a transient response, two choices must be made. First, some form of mass matrix must be decided upon. Either the consistent mass matrix prescribed by the finite element method can be employed or some form of diagonal mass matrix may be introduced. Secondly, some particular time integration procedure must be adopted. The procedures available divide themselves into two classes: the conditionally stable explicit schemes and the unconditionally or conditionally stable implicit schemes. The choices should be guided by both economy and accuracy. Using exact discrete solutions compared to the exact solutions of the differential equations, the results of these choices are displayed. Concrete examples of well-matched methods, as well as ill-matched methods, are identified and demonstrated. In particular, the diagonal mass matrix and the explicit central difference time integration method are shown to be a good combination in terms of accuracy and economy.     

Weighting parameters for time‐step integration algorithms with predetermined coefficients

In this paper, the effect of using the predetermined coefficients in constructing time‐step integration algorithms is investigated. Both first‐ and second‐order equations are considered. The approximate solution is assumed to be in a form of polynomial in the time domain. It can be related to the truncated Taylor's series expansion of the exact solution. Therefore, some of the coefficients can be predetermined from the known initial conditions. If there are m predetermined coefficients and r unknown coefficients in the approximate solution, the unknowns can be solved by the weighted residual method. The weighting parameter method is used to investigate the resultant algorithm characteristics. It is shown that the formulation is consistent with a minimum order of accuracy m+r. The maximum order of accuracy achievable is m+2r. Unconditionally stable algorithms exist if m⩽r for first‐order equations and m+1⩽r for second‐order equations. Hence, the Dahlquist's theorem is generalized. Algorithms equivalent to the Padé approximations and unconditionally stable algorithms equivalent to the generalized Padé approximations are constructed. The corresponding weighting parameters and weighting functions for the Padé and generalized Padé approximations are given explicitly. Copyright © 2000 John Wiley & Sons, Ltd.     

Finite element formulation of exact non‐reflecting boundary conditions for the time‐dependent wave equation

A modified version of an exact Non‐reflecting Boundary Condition (NRBC) first derived by Grote and Keller is implemented in a finite element formulation for the scalar wave equation. The NRBC annihilate the first N wave harmonics on a spherical truncation boundary, and may be viewed as an extension of the second‐order local boundary condition derived by Bayliss and Turkel. Two alternative finite element formulations are given. In the first, the boundary operator is implemented directly as a ‘natural’ boundary condition in the weak form of the initial–boundary value problem. In the second, the operator is implemented indirectly by introducing auxiliary variables on the truncation boundary. Several versions of implicit and explicit time‐integration schemes are presented for solution of the finite element semidiscrete equations concurrently with the first‐order differential equations associated with the NRBC and an auxiliary variable. Numerical studies are performed to assess the accuracy and convergence properties of the NRBC when implemented in the finite element method. The results demonstrate that the finite element formulation of the (modified) NRBC is remarkably robust, and highly accurate. Copyright © 1999 John Wiley & Sons, Ltd.     

Exact finite element solutions to the Helmholtz equation

For one-, two- and three-dimensional co-ordinate systems finite element matrices for the wave or Helmholtz equation are used to produce a single difference equation holding at any point of a regular mesh. Under homogeneous Dirichlet or Neumann boundary conditions, these equations are solved exactly. The eigenfunctions are the discrete form of sine or cosine functions and the eigenvalues are shown to be in error by a term of + O(h²ⁿ) where n is the order of the polynomial approximation of the wave function. The solutions provide the means of testing computer programs against the exact solutions and allow comparison with other difference schemes.     

A finite element method for non-Fourier heat conduction in strong thermal shock environments

Non-Fourier effect is important in heat conduction in strong thermal environments. Currently, generally-purposed commercial finite element code for non-Fourier heat conduction is not available. In this paper, we develop a finite element code based on a hyperbolic heat conduction equation, which includes the non-Fourier effect in heat conduction. The finite element space discretization is used to obtain a system of differential equations for the time. The transient responses are obtained by solving the system of differential equations, based on the finite difference, mode superposition, or exact time integral. The code is validated by comparing the numerical results with exact solutions for some special cases. The stability analysis is conducted and it shows that the finite difference scheme is an ideal method for the transient solution of the temperature field. It is found that with mesh refining (decreasing mesh size) and/or high-order elements, the oscillation in the vicinity of sharp change vanishes, and can be essentially suppressed by the finite difference scheme. A relationship between the time step and the space length of the element was identified to ensure that numerical oscillation vanishes.     

A finite element method for non-Fourier heat conduction in strong thermal shock environments

Non-Fourier effect is important in heat conduction in strong thermal environments. Currently, generally-purposed commercial finite element code for non-Fourier heat conduction is not available. In this paper, we develop a finite element code based on a hyperbolic heat conduction equation, which includes the non-Fourier effect in heat conduction. The finite element space discretization is used to obtain a system of differential equations for the time. The transient responses are obtained by solving the system of differential equations, based on the finite difference, mode superposition, or exact time integral. The code is validated by comparing the numerical results with exact solutions for some special cases. The stability analysis is conducted and it shows that the finite difference scheme is an ideal method for the transient solution of the temperature field. It is found that with mesh refining (decreasing mesh size) and/or high-order elements, the oscillation in the vicinity of sharp change vanishes, and can be essentially suppressed by the finite difference scheme. A relationship between the time step and the space length of the element was identified to ensure that numerical oscillation vanishes.     

Temporal‐spatial dispersion and stability analysis of finite element method in explicit elastodynamics

This work presents the temporal‐spatial (full) dispersion and stability analysis of plane square linear and biquadratic serendipity finite elements in explicit numerical solution of transient elastodynamic problems. Here, the central difference method, as an explicit time integrator, is exploited. The paper complements and extends the previous work on spatial/grid dispersion analysis of plane square biquadratic serendipity finite elements. We report on a computational strategy for temporal‐spatial dispersion relationships, where eigenfrequencies from grid/spatial dispersion analysis are adjusted to comply with the time integration method. Besides that, an ‘optimal’ lumped mass matrix for the studied finite element types is proposed and investigated. Based on the temporal‐spatial dispersion and stability analysis, relationships suggesting the ‘proper’ choice of mesh size and time step size from knowledge of the loading spectrum are presented. Copyright © 2015 John Wiley & Sons, Ltd.     

Transient behaviour of branched polymer melts through planar abrupt and rounded contractions using pom–pom models

In this paper, we study the transient flow of branched polymer melts with contrasting shear and elongational properties in planar 4:1 abrupt and rounded-corner contractions. This includes Single and Double Extended forms of the Pom–Pom model (SXPP and DXPP), comparing the transient behaviour for these two different models. With the DXPP version, the evolution of the molecular-chain backbone stretch (λ) is described by a dynamic equation, whilst in the SXPP form, stretch is an instantaneous algebraic function of the stress tensor (τ). Simulations are performed with a hybrid finite volume/element algorithm. The momentum and continuity equations are solved by a Taylor–Galerkin/pressure-correction finite element method, whilst the constitutive equation is dealt with by a cell-vertex finite volume algorithm. We demonstrate some novel features due to the influence and imposition of realistic transient boundary conditions on evolutionary flow-structure. The different effects of various model parameter choices are also exposed through transient field response in principle stress difference fringe patterns, rates of deformation, first and second normal stress difference, stress and stretch.     

Precorrected FFT accelerated BEM for large‐scale transient elastodynamic analysis using frequency‐domain approach

A precorrected fast Fourier transform (pFFT) accelerated boundary element method (BEM) for large‐scale transient elastodynamic analysis is developed and described in this paper. The frequency‐domain approach is used. To overcome the ‘wrap‐around’ problem associated with the discrete Fourier transform, the exponential window method (EWM) is employed and incorporated in the frequency‐domain BEM. An improved implementation scheme of the pFFT method based on polynomial interpolation technique is developed and applied to accelerate the elastodynamic BEM. This new scheme reduces the memory required to save the convolution matrix by a factor of 8. To further improve the efficiency of the code, a newly developed linear system solver based on the induced dimension reduction method is employed. Its performance is investigated and compared with that of the well‐known GMRES. The accuracy and computational efficiency of the method are evaluated and demonstrated by three examples: a classical benchmark, a plate subject to an impact loading and a porous cube with nearly half million DOFs subject to a step traction loading. Both analytical and experimental results are employed to validate the method. It has been found that the EWM can effectively resolve the wrap‐around problem and accurate time responses for an arbitrarily chosen time period can be obtained. Copyright © 2011 John Wiley & Sons, Ltd.     

Variations on the fundamental principle for linear systems of partial differential and difference equations with constant coefficients

New and known spaces of locally finite or polynomial exponential multivariate sequences and functions are constructed by means of substantial theorems from Commutative Algebra. They satisfy Ehrenpreis'fundamental principle and hence permit the solution of linear systems of partial differential or difference equations with constant coefficients. On the one hand this paper thus continues the author's work on multidimensional linear systems, on the other hand it generalizes and improves related work in approximation theory.     

Investigation of heat transfer coefficient in two‐dimensional transient inverse heat conduction problems using the hybrid inverse scheme

A hybrid scheme of the Laplace transform, finite difference and least‐squares methods in conjunction with a sequential‐in‐time concept, cubic spline and temperature measurements is applied to predict the heat transfer coefficient distribution on a boundary surface in two‐dimensional transient inverse heat conduction problems. In this study, the functional form of the heat transfer coefficient is unknown a priori. The whole spatial domain of the unknown heat transfer coefficient is divided into several analysis sub‐intervals. Later, a series of connected cubic polynomial function in space and a linear function in time can be applied to estimate the unknown surface conditions. Due to the application of the Laplace transform, the unknown heat transfer coefficient can be estimated from a specific time. In order to evidence the accuracy of the present inverse scheme, comparisons among the present estimates, previous results and exact solution are made. The results show that the present inverse scheme not only can reduce the number of the measurement locations but also can increase the accuracy of the estimated results. Good estimation on the heat transfer coefficient can be obtained from the knowledge of the transient temperature recordings even in the case with measurement errors. Copyright © 2007 John Wiley & Sons, Ltd.     

Conservation laws for one-dimensional isentropic gas flows

The governing equations of one-dimensional isentropic gas flow are expressed in terms of a pair of exterior differential forms. By employing Cartan's theory and the classical Frobenius theorem linear partial differential equations are obtained to determine conservation laws which are conjectured to be the key to detect completely integrable systems. By using a similarity technique explicit expressions are provided for polynomial type of conservation laws in terms of Gegenbauer and Chebyshev polynomials.     

The curved beam/deep arch/finite ring element revisited

In this paper, an attempt is made to understand the errors arising in curved finite elements which undergo both flexural and membrane deformations. It is shown that with elements of finite size (i.e. a practical level of discretization at which reasonably accurate results can be expected), there can be errors of a special nature that arise because the membrane strain fields are not consistently interpolated with terms from the two independent field functions that characterize such a problem. These lead to errors, described here as of the ‘second kind’ and a physical phenomenon called ‘membrane locking’. The findings here emerge from recent research on the effect of reduced integration on shallow curved beam elements and on the use of coupled displacement fields in finite rings. The failures which have occurred in earlier attempts to use independent polynomial displacement fields for curved elements may not have been due to neglect of rigid body motions or failure to achieve constant strain states, but because of locking due to spurious constraints. These emerge in the penalty limits of extreme thinness (an inextensional regime), when exact integration of the energy functional of an element based on low order independent interpolations for the in-plane and normal displacements is used. It seems possible to determine optimal integration rules that will allow the extensional deformation of a curved beam/deep arch/finite ring element to be modelled by independently chosen low order polynomial functions and which will recover the inextensional case in the penalty limit of extreme thinness without spurious locking constraints. The much maligned ‘cubic in w–lincar in u’ curved beam element is now reworked to show its excellent behaviour in all situations. What is emphasized is that the choice of shape functions, or subsequent operations to determine the discretized functionals, must consistently model the physical requirements the problem imposes on the field variables. In this manner, we can restore an old element to respectability and thereby indicate clearly the underlying principles. These are: the importance of ‘field consistency’ so that arch and shell problems can be modelled consistently by independent polynomial displacement fields, and the role that reduced integration or some equivalent construction can play to achieve this.     

A novel non‐linearly explicit second‐order accurate L‐stable methodology for finite deformation: hypoelastic/hypoelasto‐plastic structural dynamics problems

A novel non‐linearly explicit second‐order accurate L‐stable computational methodology for integrating the non‐linear equations of motion without non‐linear iterations during each time step, and the underlying implementation procedure is described. Emphasis is placed on illustrative non‐linear structural dynamics problems employing both total/updated Lagrangian formulations to handle finite deformation hypoelasticity/hypoelasto‐plasticity models in conjunction with a new explicit exact integration procedure for a particular rate form constitutive equation. Illustrative numerical examples are shown to demonstrate the robustness of the overall developments for non‐linear structural dynamics applications. Copyright © 2004 John Wiley & Sons, Ltd.     

求解变系数对流扩散反应方程的指数型高精度紧致差分方法

本文给出了一种数值求解变系数对流扩散反应方程的指数型高精度紧致差分方法.我们首先将模型方程变形,借助常系数对流扩散方程的指数型高精度紧致差分格式,采用残量修正法得到变系数对流扩散反应方程的指数型高精度紧致差分格式;并从理论上分析了当Pelect数很大时,本文格式达到四阶计算精度时网格步长的限制条件;离散得到的代数方程组可采用追赶法直接求解.数值实验结果与理论分析完全吻合,表明了本文格式对于边界层问题或大梯度变化的物理量求解问题具有的高精度和鲁棒性的优点.     

A closed form solution for the G/M/r machine interference model

This paper solves the machine interference problem in which N different machines are looked after by a team of r operatives. The run time of each machine is assumed to have a general distribution, different for each machine and the repair times are assumed to have a negative exponential distribution with different means for the different machines. An explicit expression for the probability that a particular group of machines is found running in the steady state is derived. From this other useful measures for the system can be obtained. It is shown that these depend on the run time distributions only through the means of those distributions.     

A residual a posteriori error estimate for partition of unity finite elements for three-dimensional transient heat diffusion problems using multiple global enrichment functions

In this article, a study of residual based a posteriori error estimation is presented for the partition of unity finite element method (PUFEM) for three-dimensional (3D) transient heat diffusion problems. The proposed error estimate is independent of the heuristically selected enrichment functions and provides a useful and reliable upper bound for the discretization errors of the PUFEM solutions. Numerical results show that the presented error estimate efficiently captures the effect of h-refinement and q-refinement on the performance of PUFEM solutions. It also efficiently reflects the effect of ill-conditioning of the stiffness matrix that is typically experienced in the partition of unity based finite element methods. For a problem with a known exact solution, the error estimate is shown to capture the same solution trends as obtained by the classical L₂ norm error. For problems with no known analytical solutions, the proposed estimate is shown to be used as a reliable and efficient tool to predict the numerical errors in the PUFEM solutions of 3D transient heat diffusion problems.     

A Fourier analysis and dynamic optimization of the Petrov–Galerkin finite element method

A Fourier analysis of the linear and quadratic N + 1 and N + 2 Petrov–Galerkin finite element methods applied to the one-dimensional transient convective-diffusion equation is performed. The results show that a priori optimization of the N + 1 method is not possible because dissipative errors are introduced as dispersive errors are reduced (any optimization is subjective). However, a priori optimization of the N + 2 Petrov–Galerkin method is possible because the reduction of dispersion errors can be accomplished without the addition of artificial dissipation. The Spectrally Weighted Average Phase Error Method (SWAPEM) for the optimization of the N + 2 Petrov–Galerkin method is introduced, in which the N + 2 weighting parameter is chosen at each time step to minimize the integral over wave number of the phase error of Fourier modes, weighted by the frequency content of the global solution at the previous time step (obtained via FFT). The method is dynamic, and general in that the dependence of the weighting parameter on the solution waveform is accounted for. Optimal values predicted by the method are in excellent agreement with those suggested by the numerical experimentation of others. Simulations of the pure convective transport of a Gaussian plume and a triangle wave are discussed to illustrate the effectiveness of the method.