Weighting parameters for unconditionally stable higher‐order accurate time step integration algorithms. Part 1—first‐order equations

In this paper, unconditionally stable higher‐order accurate time step integration algorithms suitable for linear first‐order differential equations based on the weighted residual method are presented. Instead of specifying the weighting functions, the weighting parameters are used to control the algorithm characteristics. If the numerical solution is approximated by a polynomial of degree n, the approximation is at least nth‐order accurate. By choosing the weighting parameters carefully, the order of accuracy can be improved. The generalized Padé approximations with polynomials of degree n as the numerator and denominator are considered. The weighting parameters are chosen to reproduce the generalized Padé approximations. Once the weighting parameters are known, any set of linearly independent basic functions can be used to construct the corresponding weighting functions. The stabilizing weighting factions for the weighted residual method are then found explicitly. The accuracy of the particular solution due to excitation is also considered. It is shown that additional weighting parameters may be required to maintain the overall accuracy. The corresponding equations are listed and the additional weighting parameters are solved explicitly. However, it is found that some weighting functions could satisfy the listed equations automatically. Copyright © 1999 John Wiley & Sons, Ltd.     

Solving initial value problems by differential quadrature method—part 1: first‐order equations

In this paper, the differential quadrature method is used to solve first‐order initial value problems. The initial condition is given at the beginning of a time interval. The time derivative at a sampling grid point within the time interval can be expressed as a weighted linear sum of the given initial condition and the function values at the sampling grid points within the time interval. The order of accuracy and the stability property of the quadrature solutions depend on the locations of the sampling grid points. It is shown that the order of accuracy of the quadrature solutions at the end of a time interval can be improved to 2n–1 or 2n if the n sampling grid points are chosen carefully. In fact, the approximate solutions are equivalent to the generalized Padé approximations. The resultant algorithms are therefore unconditionally stable with controllable numerical dissipation. The corresponding sampling grid points are found to be given by the roots of the modified shifted Legendre polynomials. From the numerical examples, the accuracy of the quadrature solutions obtained by using the proposed sampling grid points is found to be better than those obtained by the commonly used uniformly spaced or Chebyshev–Gauss–Lobatto sampling grid points. Copyright © 2001 John Wiley & Sons, Ltd.     

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.     

On the equivalence of the time domain differential quadrature method and the dissipative Runge–Kutta collocation method

Numerical solutions for initial value problems can be evaluated accurately and efficiently by the differential quadrature method. Unconditionally stable higher order accurate time step integration algorithms can be constructed systematically from this framework. It has been observed that highly accurate numerical results can also be obtained for non‐linear problems. In this paper, it is shown that the algorithms are in fact related to the well‐established implicit Runge–Kutta methods. Through this relation, new implicit Runge–Kutta methods with controllable numerical dissipation are derived. Among them, the non‐dissipative and asymptotically annihilating algorithms correspond to the Gauss methods and the Radau IIA methods, respectively. Other dissipative algorithms between these two extreme cases are shown to be B‐stable (or algebraically stable) as well and the order of accuracy is the same as the corresponding Radau IIA method. Through the equivalence, it can be inferred that the differential quadrature method also enjoys the same stability and accuracy properties. Copyright © 2001 John Wiley & Sons, Ltd.     

Higher‐order multi‐resolution topology optimization using the finite cell method

This article presents a detailed study on the potential and limitations of performing higher‐order multi‐resolution topology optimization with the finite cell method. To circumvent stiffness overestimation in high‐contrast topologies, a length‐scale is applied on the solution using filter methods. The relations between stiffness overestimation, the analysis system, and the applied length‐scale are examined, while a high‐resolution topology is maintained. The computational cost associated with nested topology optimization is reduced significantly compared with the use of first‐order finite elements. This reduction is caused by exploiting the decoupling of density and analysis mesh, and by condensing the higher‐order modes out of the stiffness matrix. Copyright © 2016 John Wiley & Sons, Ltd.     

Transient algorithms for heat transfer: general developments and approaches for theoretically generating Nth‐order time‐accurate operators including practically useful second‐order forms

New theoretical ideas and developments describing the fundamental underlying basis for formulating a general family of time discretization operators for first‐order parabolic systems emanating from the framework of a generalized time weighted philosophy are first presented which can be broadly classified as pertaining to Type 1, Type 2 and Type 3 family of time discretization operators. As a consequence, the evolution including the clear distinction and the bridging of the relationships between time operators termed as integral operators to the so‐called integration operators in time are theoretically developed and demonstrated. The present developments seem to not only provide avenues leading to new algorithms for transient analysis but also provide generalizations and framework to recover a wide variety of existing algorithms. Consequently, under the umbrella of the present framework, a variety of plausible new approaches for generating Nth‐order accurate time discretization operators from approximations introduced to Type 1 integral operators in time are first described followed by the developments systematically leading to Type 2 time discretization operators, and subsequently to a wide class of Type 3 time integration operators including the recovery of a variety of known existing time integration operators which can be uniquely identified by Discrete Numerically Assigned (DNA) algorithmic markers. Of the various developments, of noteworthy mention and emphasis here are a new family of L‐stable Nth‐order Integration Operators (LNInO) of Type 2 for transient computations. Subsequently, some practically useful second‐order forms are specifically illustrated and highlighted. The stability and accuracy characteristics are also described for a variety of generated algorithms applicable for transient heat transfer computations. Although the primary focus is on the theoretical developments encompassing linear operators, some simple numerical examples are finally demonstrated to merely illustrate the salient features of the proposed developments. Copyright © 1999 John Wiley & Sons, Ltd.     

Higher‐order modal transformation for reduced‐order modeling of linear systems undergoing global parametric variations

Previously, a novel parametric reduced‐order model technique for linear systems was developed based on a frequency‐domain formulation and the so‐called modally equivalent perturbed system. The main advantage of the scheme is that it isolates all the perturbed matrices into a forcing term, allowing for a simple and powerful analysis based on the ordinary differential equation with the forcing input. It was shown that when the parameter variation is limited to a finite dimension, it yields exceptionally accurate reduced‐order models for a wide range of parameter values. In this paper, the original method is improved to cover a larger‐dimensional domain and the global domain of the variation by adding higher‐order terms in the formulation. It is shown that when expressed in powers of incremental matrices, the new formula resembles a well‐known series expansion. The improved parametric reduced‐order model is demonstrated for a computational fluid dynamics model of unsteady air flow around a two‐dimensional airfoil in subsonic flows with Mach variation.     

Krylov precise time‐step integration method

An efficient precise time‐step integration (PTI) algorithm to solve large‐scale transient problems is presented in this paper. The Krylov subspace method and the Padé approximations are applied to modify the original PTI algorithm in order to improve the computational efficiency. Both the stability and accuracy characteristics of the resultant algorithms are investigated. The efficiency can be further improved by expanding the dimension to avoid the computation of the particular solutions. The present algorithm can also be extended to tackle nonlinear problems without difficulty. Two numerical examples are given to illustrate the highly accurate and efficient algorithm. Copyright © 2006 John Wiley & Sons, Ltd.     

The tetrahedral finite cell method: Higher‐order immersogeometric analysis on adaptive non‐boundary‐fitted meshes

The finite cell method (FCM) is an immersed domain finite element method that combines higher‐order non‐boundary‐fitted meshes, weak enforcement of Dirichlet boundary conditions, and adaptive quadrature based on recursive subdivision. Because of its ability to improve the geometric resolution of intersected elements, it can be characterized as an immersogeometric method. In this paper, we extend the FCM, so far only used with Cartesian hexahedral elements, to higher‐order non‐boundary‐fitted tetrahedral meshes, based on a reformulation of the octree‐based subdivision algorithm for tetrahedral elements. We show that the resulting TetFCM scheme is fully accurate in an immersogeometric sense, that is, the solution fields achieve optimal and exponential rates of convergence for h‐refinement and p‐refinement, if the immersed geometry is resolved with sufficient accuracy. TetFCM can leverage the natural ability of tetrahedral elements for local mesh refinement in three dimensions. Its suitability for problems with sharp gradients and highly localized features is illustrated by the immersogeometric phase‐field fracture analysis of a human femur bone. Copyright © 2016 John Wiley & Sons, Ltd.     

Step size adjustment and extrapolation for time‐stepping schemes in non‐smooth dynamics

In this paper we use step size adjustment and extrapolation methods to improve Moreau's time‐stepping scheme for the numerical integration of non‐smooth mechanical systems, i.e. systems with impact and friction. The scheme yields a system of inclusions, which is transformed into a system of projective equations. These equations are solved iteratively. Switching points are time instants for which the structure of the mechanical system changes, for example, time instants for which a sticking friction element begins to slide. We show how switching points can be localized and how these points can be resolved by choosing a minimal step size. In order to improve the integration of non‐smooth systems in the smooth parts, we show how the time‐stepping method can be used as a base integration scheme for extrapolation methods, which allow for an increase in the integration order. Switching points are processed by a small time step, while time intervals during which the structure of the system does not change are computed with a larger step size and improved integration order. The overall algorithm, which consists of a time‐stepping module, an extrapolation module and a step size adjustment module, is discussed in detail and some examples are given. Copyright © 2008 John Wiley & Sons, Ltd.     

A new unified theory underlying time dependent linear first‐order systems: a prelude to algorithms by design

A new unified theory underlying the theoretical design of linear computational algorithms in the context of time dependent first‐order systems is presented. Providing for the first time new perspectives and fresh ideas, and unlike various formulations existing in the literature, the present unified theory involves the following considerations: (i) it leads to new avenues for designing new computational algorithms to foster the notion of algorithms by design and recovering existing algorithms in the literature, (ii) describes a theory for the evolution of time operators via a unified mathematical framework, and (iii) places into context and explains/contrasts future new developments including existing designs and the various relationships among the different classes of algorithms in the literature such as linear multi‐step methods, sub‐stepping methods, Runge–Kutta type methods, higher‐order time accurate methods, etc. Subsequently, it provides design criteria and guidelines for contrasting and evaluating time dependent computational algorithms. The linear computational algorithms in the context of first‐order systems are classified as distinctly pertaining to Type 1, Type 2, and Type 3 classifications of time discretized operators. Such a distinct classification, provides for the first time, new avenues for designing new computational algorithms not existing in the literature and recovering existing algorithms of arbitrary order of time accuracy including an overall assessment of their stability and other algorithmic attributes. Consequently, it enables the evaluation and provides the relationships of computational algorithms for time dependent problems via a standardized measure based on computational effort and memory usage in terms of the resulting number of equation systems and the corresponding number of system solves. A generalized stability and accuracy limitation barrier theorem underlies the generic designs of computational algorithms with arbitrary order of accuracy and establishes guidelines which cannot be circumvented. In summary, unlike the traditional approaches and classical school of thought customarily employed in the theoretical development of computational algorithms, the unified theory underlying time dependent first‐order systems serves as a viable avenue to foster the notion of algorithms by design. Copyright © 2004 John Wiley & Sons, Ltd.     

Higher‐order boundary element methods for transient diffusion problems. Part II: Singular flux formulation

A boundary element method (BEM) for transient heat diffusion phenomena presented in Part I is extended to problems involving instantaneous rise of temperature on a portion of the boundary. The new boundary element formulation involves the use of an infinite flux function in order to properly capture the singular response of the flux. It is shown that the conventional finite flux BEM formulation, as well as a commercial FEM code, results in a large first‐time‐step numerical error that cannot be reduced by mesh or time‐step refinement. The use of the singular flux formulation for BEM demonstrates an extremely high level of accuracy for the one‐dimensional case, and a significant improvement in the solutions within a two‐dimensional representation. The additional errors arising due to improper time interpolation of the temperature on the boundaries adjacent to the singular flux boundary are discussed. Copyright © 2002 John Wiley & Sons, Ltd.     

Multi‐time‐step explicit–implicit method for non‐linear structural dynamics

We present a method with domain decomposition to solve time‐dependent non‐linear problems. This method enables arbitrary numeric schemes of the Newmark family to be coupled with different time steps in each subdomain: this coupling is achieved by prescribing continuity of velocities at the interface. We are more specifically interested in the coupling of implicit/explicit numeric schemes taking into account material and geometric non‐linearities. The interfaces are modelled using a dual Schur formulation where the Lagrange multipliers represent the interfacial forces. Unlike the continuous formulation, the discretized formulation of the dynamic problem is unable to verify simultaneously the continuity of displacements, velocities and accelerations at the interfaces. We show that, within the framework of the Newmark family of numeric schemes, continuity of velocities at the interfaces enables the definition of an algorithm which is stable for all cases envisaged. To prove this stability, we use an energy method, i.e. a global method over the whole time interval, in order to verify the algorithms properties. Then, we propose to extend this to non‐linear situations in the following cases: implicit linear/explicit non‐linear, explicit non‐linear/explicit non‐linear and implicit non‐linear/explicit non‐linear. Finally, we present some examples showing the feasibility of the method. Copyright © 2001 John Wiley & Sons, Ltd.     

Design of order‐preserving algorithms for transient first‐order systems with controllable numerical dissipation

Using a new design procedure termed as Algorithms by Design, which we have successfully introduced in our previous efforts for second‐order systems, alternatively, we advance in this exposition, the design and development of a computational framework that permits order‐preserving second‐order time accurate, unconditionally stable, zero‐order overshooting behavior, and features with controllable numerical dissipation and dispersion via a family of algorithms for effectively solving transient first‐order systems. The key feature is the incorporation of a spurious root to introduce controllable numerical dissipation while preserving second‐order accuracy (order‐preserving feature) resulting in a two‐root system, namely, the principal root (ρ_1∞) and a spurious root (ρ_2∞). In contrast to the classical Trapezoidal family of algorithms which are the most popular, the present framework has the same order of computational complexity, but a higher payoff that is a significant advance to the field for tackling a wide class of applications dealing with first‐order transient systems. We also present the special case with selection of ρ_1∞ = 1 and any ρ_2∞ leading to the design of a family of generalized single‐step single‐solve [GS4‐1] algorithms recovering the Crank–Nicolson method at one end (ρ_2∞ = 1) and the Midpoint Rule at the other end (ρ_2∞ = 0) and anything in between, all of which have spectral radius features resembling that of the Crank–Nicolson method. More interestingly, with the particular choice of ρ_1∞ = ρ_2∞ = 0, the developed framework additionally inherits L‐stable features. We illustrate the successful design of the developed GS4‐1 framework using two simple illustrative numerical examples. Copyright © 2011 John Wiley & Sons, Ltd.     

Numerical solution of second‐order,two‐point boundary value problems using continuous genetic algorithms

Second‐order, two‐point boundary‐value problems are encountered in many engineering applications including the study of beam deflections, heat flow, and various dynamic systems. Two classical numerical techniques are widely used in the engineering community for the solution of such problems; the shooting method and finite difference method. These methods are suited for linear problems. However, when solving the non‐linear problems, these methods require some major modifications that include the use of some root‐finding technique. Furthermore, they require the use of other basic numerical techniques in order to obtain the solution. In this paper, the author introduces a novel method based on continuous genetic algorithms for numerically approximating a solution to this problem. The new method has the following characteristics; first, it does not require any modification while switching from the linear to the non‐linear case; as a result, it is of versatile nature. Second, this approach does not resort to more advanced mathematical tools and is thus easily accepted in the engineering application field. Third, the proposed methodology has an implicit parallel nature which points to its implementation on parallel machines. However, being a variant of the finite difference scheme with truncation error of the order O(h²), the method provides solutions with moderate accuracy. Numerical examples presented in the paper illustrate the applicability and generality of the proposed method. Copyright © 2004 John Wiley & Sons, Ltd.     

Accuracy of one‐step integration schemes for damped/forced linear structural dynamics

This paper proposes an energy‐based measure for the evaluation of the local truncation error of two‐level one‐step integration schemes. The measure applies to multiple degree of freedom systems and does not necessarily require modal reduction to a scalar model; it naturally handles the structural damping and external forcing terms that are generally and mistakenly neglected in error analyses, and it segregates the error associated with the free and forced response components of the problem. To illustrate the approach, two examples associated with the application of the trapezoidal scheme and of a high‐order scheme proposed in the literature are analyzed. The latter reveals the shortcomings of the standard approach that is based on the undamped/unforced linear oscillator and therefore highlights the need for the proposed framework. Indeed, the scheme order of accuracy is below expectation when structural damping or external forcing is considered, in the numerically dissipative setting. Developments on the basis of the time discontinuous Galerkin (TDG) method are then proposed to recover the scheme high‐order accuracy. Additionally, they show the similarity that exists between schemes related to the TDG method and the ones obtained by integration by parts of the equation of motion. Copyright © 2014 John Wiley & Sons, Ltd.     

Higher‐order responses of three‐dimensional elastic plate structures and their numerical illustration by p‐FEM

The displacements of three‐dimensional linearly elastic plate domains can be expanded as a compound power‐series asymptotics, when the thickness parameter ε tends to zero. The leading term u ⁰ in this expansion is the well‐known Kirchhoff–Love displacement field, which is the solution to the limit case when ε→0. Herein, we focus our discussion on plate domains with either clamped or free lateral boundary conditions, and characterize the loading conditions for which the leading term vanishes. In these situations the first non‐zero term u ^k in the expansion appears for k=2, 3 or 4 and is denoted as higher‐order response of order 2,3 or 4, respectively. We provide herein explicit loading conditions under which higher order responses in three‐dimensional plate structures are visible, and demonstrate the mathematical analysis by numerical simulation using the p‐version finite element method. Owing to the need for highly accurate results and ‘needle elements’ (having extremely large aspect ratio up to 10000), a p‐version finite element analysis is mandatory for obtaining reliable and highly accurate results. Copyright © 2001 John Wiley & Sons, Ltd.     

A FETI‐based domain decomposition technique for time‐dependent first‐order systems based on a DAE approach

We present a novel partitioned coupling algorithm to solve first‐order time‐dependent non‐linear problems (e.g. transient heat conduction). The spatial domain is partitioned into a set of totally disconnected subdomains. The continuity conditions at the interface are modeled using a dual Schur formulation where the Lagrange multipliers represent the interface fluxes (or the reaction forces) that are required to maintain the continuity conditions. The interface equations along with the subdomain equations lead to a system of differential algebraic equations (DAEs). For the resulting equations a numerical algorithm is developed, which includes choosing appropriate constraint stabilization techniques. The algorithm first solves for the interface Lagrange multipliers, which are subsequently used to advance the solution in the subdomains. The proposed coupling algorithm enables arbitrary numeric schemes to be coupled with different time steps (i.e. it allows subcycling) in each subdomain. This implies that existing software and numerical techniques can be used to solve each subdomain separately. The coupling algorithm can also be applied to multiple subdomains and is suitable for parallel computers. We present examples showing the feasibility of the proposed coupling algorithm. Copyright © 2008 John Wiley & Sons, Ltd.     

A semi‐Lagrangean time‐integration approach for extended finite element methods

Many computational problems incorporate discontinuities that evolve in time. The eXtendend Finite Element Method (XFEM) is able to represent discontinuities sharply on fixed arbitrary meshes, but numerical difficulties arise if these discontinuities move in time. We point out that this issue is crucial for interface problems with strongly discontinuous fields on fixed grids. A method using semi‐Lagrangean techniques is proposed to adequately handle time integration based on finite difference schemes in the context of the XFEM. The basic idea is to adapt previous numerical solutions to the current interface position by tracking back virtual Lagrangean particles to their previous positions, where an appropriate solution can be extrapolated from a smooth field. Convergence properties of the proposed method in time and space are thoroughly studied for two one‐dimensional model problems. Finally, the method is applied to the particularly challenging problem of premixed combustion, where the discontinuity appears at the flame front separating the burnt from the unburnt gases. A two‐dimensional and a three‐dimensional expanding flame demonstrates that the method is sufficiently accurate to retain the properties of the overall Nitsche‐type formulation for interface problems with embedded strong discontinuities. Copyright © 2014 John Wiley & Sons, Ltd.     

Fourth‐order energy‐preserving locally implicit time discretization for linear wave equations

A family of fourth‐order coupled implicit–explicit time schemes is presented as a special case of fourth‐order coupled implicit schemes for linear wave equations. The domain of interest is decomposed into several regions where different fourth‐order time discretizations are used, chosen among a family of implicit or explicit fourth‐order schemes. The coupling is based on a Lagrangian formulation on the boundaries between the several non‐conforming meshes of the regions. A global discrete energy is shown to be preserved and leads to global fourth‐order consistency in time. Numerical results in 1D and 2D for the acoustic and elastodynamics equations illustrate the good behavior of the schemes and their potential for the simulation of realistic highly heterogeneous media or strongly refined geometries, for which using everywhere an explicit scheme can be extremely penalizing. Accuracy up to fourth order reduces the numerical dispersion inherent to implicit methods used with a large time step and makes this family of schemes attractive compared with second‐order accurate methods. Copyright © 2015 John Wiley & Sons, Ltd.