Linear finite elements with orthogonal discontinuous basis functions for explicit earthquake ground motion modeling

The dynamic explicit finite element method is commonly used in earthquake ground motion modeling. In this method, the element mass matrix is approximately lumped, which may lead to numerical dispersion. On the other hand, the orthogonal finite element method, based on orthogonal polynomial basis functions, naturally derives a lumped diagonal mass matrix and can be applied to dynamic explicit finite element analysis. In this paper, we propose finite elements based on orthogonal discontinuous basis functions, the element mass matrices of which are lumped without approximation. Orthogonal discontinuous basis functions are used to improve the accuracy and reduce the numerical dispersion in earthquake ground motion modeling. We present a detailed formulation of the 4‐node tetrahedral and 8‐node hexahedral elements. The relationship between the proposed finite elements and conventional finite elements is investigated, and the solutions obtained from the conventional explicit finite element method are compared with analytical solutions to verify the numerical dispersion caused by the lumping approximation. Comparison of solutions obtained with the proposed finite elements to analytical solutions demonstrates the usefulness of the technique. Examples are also presented to illustrate the effectiveness of the proposed method in earthquake ground motion modeling in the actual three‐dimensional crust structure. Copyright © 2010 John Wiley & Sons, Ltd.     

Transient two-dimensional heat conduction problems solved by the finite element method

A finite element weighted residual process has been used to solve transient linear and non-linear two-dimensional heat conduction problems. Rectangular prisms in a space-time domain were used as the finite elements. The weighting function was equal to the shape function defining the dependent variable approximation. The results are compared in tables with analytical, as well as other numerical data. The finite element method compared favourably with these results. It was found to be stable, convergent to the exact solution, easily programmed, and computationally fast. Finally, the method does not require constant parameters over the entire solution domain.     

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.     

Educating local radial basis functions using the highest gradient of interest in three dimensional geometries

We present a novel methodology to effectively localize radial basis function approximation methods in three dimensions. The local scheme requires shape parameter‐dependent functions that can be used to approximate gradients of scattered data and to solve partial differential operators. The optimum shape parameter is obtained from the highest gradient of interest, where a known analytical function, when boundary conditions are not present, or a shape parameter‐free global approximation are used to educate the localized scheme. The later option is applicable to problems where the operator needs to be solved multiple times, like in time evolution or stochastic integration. Past shape parameter's optimizations, for two‐dimensional domains, based on the condition number of the interpolant matrix, were unable to provide satisfactory approximations. The applicability of our method is illustrated in the context of an analytical expression interpolation and during a Ginzburg–Landau relaxation of a free energy functional. In general, the optimum shape parameter depends on geometry, node distribution, and density, whereas the approximation errors decrease as the node density and the local stencil size increase. The effective localization of radial basis functions motivates its use in moving boundary problems and accelerates solutions through sparse matrix solvers. Copyright © 2016 John Wiley & Sons, Ltd.     

Modelling and process optimization for functionally graded materials

We optimize continuous quench process parameters to produce functionally graded aluminium alloy extrudates. To perform this task, an optimization problem is defined and solved using a standard non‐linear programming algorithm. Ingredients of this algorithm include (1) the process parameters to be optimized, (2) a cost function: the weighted average of the precipitate number density distribution, (3) constraint functions to limit the temperature gradient (and hence distortion and residual stress) and exit temperature, and (4) their sensitivities with respect to the process parameters. The cost and constraint functions are dependent on the temperature and precipitate size which are obtained by balancing energy to determine the temperature distribution and by using a reaction‐rate theory to determine the precipitate particle sizes and their distributions. Both the temperature and the precipitate models are solved via the discontinuous Galerkin finite element method. The energy balance incorporates non‐linear boundary conditions and material properties. The temperature field is then used in the reaction rate model which has as many as 10⁵ degrees‐of‐freedom per finite element node. After computing the temperature and precipitate size distributions we must compute their sensitivities. This seemingly intractable computational task is resolved thanks to the discontinuous Galerkin finite element formulation and the direct differentiation sensitivity method. A three‐dimension example is provided to demonstrate the algorithm. Copyright © 2004 John Wiley & Sons, Ltd.     

Unification of parametric and implicit methods for shape sensitivity analysis and optimization with fixed mesh

Parametric and implicit methods are traditionally thought to be two irrelevant approaches in structural shape optimization. Parametric method works as a Lagrangian approach and often uses the parametric boundary representation (B‐rep) of curves/surfaces, for example, Bezier and B‐splines in combination with the conformal mesh of a finite element model, while implicit method relies upon level‐set functions, that is, implicit functions for B‐rep, and works as an Eulerian approach in combination with the fixed mesh within the scope of extended finite element method or finite cell method. The original contribution of this work is the unification of both methods. First, a new shape optimization method is proposed by combining the features of the parametric and implicit B‐reps. Shape changes of the structural boundary are governed by parametric B‐rep on the fixed mesh to maintain the merit in computer‐aided design modeling and avoid laborious remeshing. Second, analytical shape design sensitivity is formulated for the parametric B‐rep in the framework of fixed mesh of finite cell method by means of the Hamilton–Jacobi equation. Numerical examples are solved to illustrate the unified methodology. Copyright © 2016 John Wiley & Sons, Ltd.     

A modified element‐free Galerkin method with essential boundary conditions enforced by an extended partition of unity finite element weight function

In this work we propose a method which combines the element‐free Galerkin (EFG) with an extended partition of unity finite element method (PUFEM), that is able to enforce, in some limiting sense, the essential boundary conditions as done in the finite element method (FEM). The proposed extended PUFEM is based on the moving least square approximation (MLSA) and is capable of overcoming singularity problems, in the global shape functions, resulting from the consideration of linear and higher order base functions. With the objective of avoiding the presence of singular points, the extended PUFEM considers an extension of the support of the classical PUFE weight function. Since the extended PUFEM is closely related to the EFG method there is no need for special approximation functions with complex implementation procedures, and no use of the penalty and/or multiplier method is required in order to approximately impose the essential boundary condition. Thus, a relatively simple procedure is needed to combine both methods. In order to attest the performance of the method we consider the solution of an analytical elastic problem and also some coupled elastoplastic‐damage problems. Copyright © 2003 John Wiley & Sons, Ltd.     

Dynamic response of prismatic metallic sandwich tubes under combined internal shock pressure and thermal load

This paper presents an analytical model for the transient thermomechanical response in the sandwich tube subjected to internal shock pressure and thermal load. A sandwich model (3-layers model) and a multilayer model for predicting transient temperature are proposed, and the effective thermal conductivities in each layer are obtained by the analytical method based on the Fourier law of conduction. The analysis of dynamic response is conducted by using the high order sandwich shell models, considering the shear deformation and compressibility of the core. Moreover, several numerical simulations are carried out by finite element (FE) analysis, which are compared with the results obtained by using analytical models. The slight discrepancies between the numerical simulations and analytical results indicate that the sandwich model (3-layers model) can predict the transient temperature in the sandwich tube correctly, and the thermomechanical response obtained by the high order sandwich shell model is within acceptable accuracy. Thermal stresses are strongly dependent on temperature gradient in the sandwich wall. The dynamic resultant forces with large amplitudes will induce structural fatigue, which is harmful to the structural life and reliability. Therefore, the thermal stresses should be considered for the design of sandwich tubes or pipes subjected to thermomechanical load.     

A methodology for fast finite element modeling of electrostatically actuated MEMS

In this paper, a methodology is proposed for expediting the coupled electro‐mechanical finite element modeling of electrostatically actuated MEMS. The proposed methodology eliminates the need for repeated finite element meshing and subsequent electrostatic modeling of the device during mechanical deformation. We achieve this by using an approximation of the charge density on the movable electrode in the deformed geometry in terms of the charge density in the non‐deformed geometry and displacements of the movable electrode. The electrostatic problem has to be solved only once and thus this method speeds up the coupled electro‐mechanical simulation process. The proposed methodology is demonstrated through its application to the modeling of four MEMS devices with varying length‐to‐gap ratios, multiple dielectrics and complicated geometries. Its accuracy is assessed through comparisons of its results with results obtained using both analytical solutions and finite element solutions obtained using ANSYS. Copyright © 2008 John Wiley & Sons, Ltd.     

Dynamic response of prismatic metallic sandwich tubes under combined internal shock pressure and thermal load

This paper presents an analytical model for the transient thermomechanical response in the sandwich tube subjected to internal shock pressure and thermal load. A sandwich model (3-layers model) and a multilayer model for predicting transient temperature are proposed, and the effective thermal conductivities in each layer are obtained by the analytical method based on the Fourier law of conduction. The analysis of dynamic response is conducted by using the high order sandwich shell models, considering the shear deformation and compressibility of the core. Moreover, several numerical simulations are carried out by finite element (FE) analysis, which are compared with the results obtained by using analytical models. The slight discrepancies between the numerical simulations and analytical results indicate that the sandwich model (3-layers model) can predict the transient temperature in the sandwich tube correctly, and the thermomechanical response obtained by the high order sandwich shell model is within acceptable accuracy. Thermal stresses are strongly dependent on temperature gradient in the sandwich wall. The dynamic resultant forces with large amplitudes will induce structural fatigue, which is harmful to the structural life and reliability. Therefore, the thermal stresses should be considered for the design of sandwich tubes or pipes subjected to thermomechanical load.     

A SPACE-TIME COUPLED p-VERSION LEAST SQUARES FINITE ELEMENT FORMULATION FOR UNSTEADY TWO-DIMENSIONAL NAVIER–STOKES EQUATIONS

This paper presents a p-version least squares finite element formulation for two-dimensional unsteady fluid flow described by Navier–Stokes equations where the effects of space and time are coupled. The dimensionless form of the Navier–Stokes equations are first cast into a set of first-order differential equations by introducing auxiliary variables. This permits the use of C⁰ element approximation. The element properties are derived by utilizing the p-version approximation functions in both space and time and then minimizing the error functional given by the space–time integral of the sum of squares of the errors resulting from the set of first-order differential equations. This results in a true space–time coupled least squares minimization procedure. The application of least squares minimization to the set of coupled first-order partial differential equations results in finding a solution vector {δ} which makes gradient of error functional with respect to {δ} a null vector. This is accomplished by using Newton's method with a line search. A time marching procedure is developed in which the solution for the current time step provides the initial conditions for the next time step. Equilibrium iterations are carried out for each time step until the error functional and each component of the gradient of the error functional with respect to nodal degrees of freedom are below a certain prespecified tolerance. The space–time coupled p-version approximation functions provide the ability to control truncation error which, in turn, permits very large time steps. What literally requires hundreds of time steps in uncoupled conventional time marching procedures can be accomplished in a single time step using the present space–time coupled approach. The generality, success and superiority of the present formulation procedure is demonstrated by presenting specific numerical examples for transient couette flow and transient lid driven cavity. The results are compared with the analytical solutions and those reported in the literature. The formulation presented here is ideally suited for space–time adaptive procedures. The element error functional values provide a mechanism for adaptive h, p or hp refinements.     

Two‐dimensional finite element method solution of a class of integro‐differential equations: Application to non‐Fickian transport in disordered media

A finite element method is developed to solve a class of integro‐differential equations and demonstrated for the important specific problem of non‐Fickian contaminant transport in disordered porous media. This transient transport equation, derived from a continuous time random walk approach, includes a memory function. An integral element is the incorporation of the well‐known sum‐of‐exponential approximation of the kernel function, which allows a simple recurrence relation rather than storage of the entire history. A two‐dimensional linear element is implemented, including a streamline upwind Petrov–Galerkin weighting scheme. The developed solver is compared with an analytical solution in the Laplace domain, transformed numerically to the time domain, followed by a concise convergence assessment. The analysis shows the power and potential of the method developed here. Copyright © 2017 John Wiley & Sons, Ltd.     

Simulation of transient heat conduction using one‐dimensional mapped infinite elements

Many engineering problems exist in physical domains that can be said to be infinitely large. A common problem in the simulation of these unbounded domains is that a balance must be met between a practically sized mesh and the accuracy of the solution. In transient applications, developing an appropriate mesh size becomes increasingly difficult as time marches forward. The concept of the infinite element was introduced and implemented for elliptic and for parabolic problems using exponential decay functions. This paper presents a different methodology for modeling transient heat conduction using a simplified mesh consisting of only two‐node, one‐dimensional infinite elements for diffusion into an unbounded domain and is shown to be applicable for multi‐dimensional problems. A brief review of infinite elements applied to static and transient problems is presented. A transient infinite element is presented in which the element length is time‐dependent such that it provides the optimal solution at each time step. The element is validated against the exact solution for constant surface heat flux into an infinite half‐space and then applied to the problem of heat loss in thermal reservoirs. The methodology presented accurately models these phenomena and presents an alternative methodology for modeling heat loss in thermal reservoirs. Copyright © 2010 John Wiley & Sons, Ltd.     

A modified finite difference sensitivity analysis method allowing remeshing in large strain path‐dependent problems

Finite difference sensitivity analysis is simple and general yet usually inefficient and inaccurate compared to the analytical sensitivity approach. Although its high computational cost is not an issue in iteratively solved problems, its inaccuracies are critical in path‐dependent problems when remeshing is required. In this case, the errors caused by parametric inversion and interpolation in variables transfer to the new mesh can be as large as the gradient components. This paper presents an efficient modified finite difference approach that allows remeshing either in path‐independent or path‐dependent problems, not being affected by the aforementioned errors. The strategy to cope with remeshing is extensive to the semi‐analytical method which, for non‐linear analyses, is shown to be a particular case of the proposed finite difference sensitivity approach. With this implementation, the finite difference, the semi‐analytical and the analytical sensitivity methods all have comparable computational costs. The perturbation of unstructured meshes is performed with an inverse power Laplacian smoothing. The low cost and the accuracy of the sensitivity fields obtained after remeshing are shown in two examples, considering shape and constitutive design variables. Copyright © 2004 John Wiley & Sons, Ltd.     

Shape optimization using a NURBS‐based interface‐enriched generalized FEM

This study presents a gradient‐based shape optimization over a fixed mesh using a non‐uniform rational B‐splines‐based interface‐enriched generalized finite element method, applicable to multi‐material structures. In the proposed method, non‐uniform rational B‐splines are used to parameterize the design geometry precisely and compactly by a small number of design variables. An analytical shape sensitivity analysis is developed to compute derivatives of the objective and constraint functions with respect to the design variables. Subtle but important new terms involve the sensitivity of shape functions and their spatial derivatives. Verification and illustrative problems are solved to demonstrate the precision and capability of the method. Copyright © 2016 John Wiley & Sons, Ltd.     

Fracture in magnetoelectroelastic materials using the extended finite element method

Static fracture analyses in two‐dimensional linear magnetoelectroelastic (MEE) solids is studied by means of the extended finite element method (X‐FEM). In the X‐FEM, crack modeling is facilitated by adding a discontinuous function and the crack‐tip asymptotic functions to the standard finite element approximation using the framework of partition of unity. In this study, media possessing fully coupled piezoelectric, piezomagnetic and magnetoelectric effects are considered. New enrichment functions for cracks in transversely isotropic MEE materials are derived, and the computation of fracture parameters using the domain form of the contour interaction integral is presented. The convergence rates in energy for topological and geometric enrichments are studied. Excellent accuracy of the proposed formulation is demonstrated on benchmark crack problems through comparisons with both analytical solutions and numerical results obtained by the dual boundary element method. Copyright © 2011 John Wiley & Sons, Ltd.     

Two kinds of tube elements for transient thermal–structural analysis of large space structures

Two kinds of thin‐walled tube elements are presented for transient thermal–structural analysis of large space structures by the finite element method. Not only the average temperature, but also the perturbation temperature in the cross‐section of the tube is considered in the present elements. These two temperatures are decoupled in the deduction about the new elements and the non‐linear analysis is restricted to solving the equations of average temperature. Therefore, the magnitude of the non‐linear analysis can be reduced by the presented method. The main difference between the two kinds of thin‐walled tube elements is in the shape functions of the temperature along the circumference of cross‐section. Corresponding to the transient temperature field, quasistatic thermo‐elastic analysis is also introduced. Three examples are shown and the effectiveness of the new elements is discussed. Copyright © 2004 John Wiley & Sons, Ltd.     

Calculation of transient temperature fields with finite elements in space and time dimensions

A program is demonstrated which apart from linear finite elements in time also includes elements with shape functions of the second and third degree. The algorithm for discretization in the time dimension is described and, using the example of a parabolic time element, the coefficients required to form the global system are given. By various test examples the efficiency of the process is examined by comparison with the customary difference method. Generally, with finite elements in time, the solution has better stability. Comparing the time required for calculation with the accuracy of the solution it would appear that in examining problems where boundary conditions are constant in time, higher order time elements are no improvement over the linear time element. However, for the purpose of reproducing periodic processes, higher order time elements offer an advantage in that one is not limited to linear variations of the boundary conditions within the element. Thus, for example, the temperature curve for parabolic variation of the surface temperature can be reproduced with close approximation by two time elements per period and a shape function of the third degree.     

Three‐dimensional non‐planar crack growth by a coupled extended finite element and fast marching method

A numerical technique for non‐planar three‐dimensional linear elastic crack growth simulations is proposed. This technique couples the extended finite element method (X‐FEM) and the fast marching method (FMM). In crack modeling using X‐FEM, the framework of partition of unity is used to enrich the standard finite element approximation by a discontinuous function and the two‐dimensional asymptotic crack‐tip displacement fields. The initial crack geometry is represented by two level set functions, and subsequently signed distance functions are used to maintain the location of the crack and to compute the enrichment functions that appear in the displacement approximation. Crack modeling is performed without the need to mesh the crack, and crack propagation is simulated without remeshing. Crack growth is conducted using FMM; unlike a level set formulation for interface capturing, no iterations nor any time step restrictions are imposed in the FMM. Planar and non‐planar quasi‐static crack growth simulations are presented to demonstrate the robustness and versatility of the proposed technique. Copyright © 2008 John Wiley & Sons, Ltd.     

Dispersion analysis and element‐free Galerkin solutions of second‐ and fourth‐order gradient‐enhanced damage models

Gradient‐dependent damage formulations incorporate higher‐order derivatives of state variables in the constitutive equations. Different formulations have been derived for this gradient enhancement, comparison of which is difficult in a finite element context due to higher‐order continuity requirements for certain formulations. On the other hand, the higher‐order continuity requirements are met naturally by element‐free Galerkin (EFG) shape functions. Thus, the EFG method provides a suitable tool for the assessment of gradient enhanced continuum models. Dispersion analyses have been carried out to compare different gradient enhanced models with the non‐local damage model. The formulation of the additional boundary conditions is addressed. Numerical examples show the objectivity with respect to the discretization and the differences between various gradient formulations with second‐ and fourth‐order derivatives. It is shown that with the same underlying internal length scale, very different results can be obtained. Copyright © 2000 John Wiley & Sons, Ltd.