Step Integration Algorithm Research Articles

A Staggered Asynchronous Step Integration Algorithm for Hybrid Finite-Element and Discrete-Element Modeling

To reduce the computational time in the fields of dynamics, we introduce herein an efficient asynchronous step integration algorithm for a coupled finite-element–discrete-element method (FEM–DEM). Domain decomposition divides the whole model into an FEM subdomain and a DEM subdomain, where each subdomain selects the appropriate time step based on the element characteristics. The method of overlapping nodes and particles at the boundaries is adopted to handle the interface coupling problem, and the subcycling method is adopted to tackle the problem of transmission and matching of boundary data. Two numerical examples are presented to demonstrate the precision of the analysis and the overall efficiency. The algorithm provides the requisite calculation accuracy and significantly shortens the computer simulation. This approach further improves the efficiency of hybrid FEM–DEM modeling.

A non-linear Ritz method for the analysis of low velocity impact induced dynamics in variable angle tow composite laminates

Variable angle tow (VAT) laminates feature composite layers reinforced by fibres following continuous curved paths and offer a wide structural design space for the manufacturing of composite components. In this work, a formulation for the analysis of the impact-induced dynamics in VAT laminated plates is proposed, implemented and tested in this work. The method is based on the adoption of first order shear deformation kinematics and includes von Karman non-linear strains. The discrete system is obtained by employing a pb-2 Ritz series expansion into the Hamilton’s variational statement, while the impact loading is modelled through Hertzian contact law. The resulting non-linear governing equations are solved using a Newmark time step integration algorithm , coupled with an iterative solution strategy. After validation against available literature data, several tests on different VAT configurations are performed, highlighting analogies and differences between different layups in terms of impact-induced dynamic response.

Numerical assessment of a double step integration algorithm for [formula omitted] plasticity with Armstrong–Frederick evolution of back stress

Abstract In this work a double step algorithm is presented for the integration of equations governing the behavior of von Mises material in plastic limit. The isotropic and kinematic hardenings employed are of general type and the evolution of back stress follows the Armstrong–Frederick rule. Theoretical and numerical aspects are discussed in detail and a comparison is made with the classical backward Euler method.

Assessment of the Computational Performances of the Semi-Analytical Method (SAM) for Computing 2-D Current Distributions in Superconductors

The semi-analytical method (SAM) is a fast integral technique for solving small 2-D, time-transient electromagnetic problems in high temperature superconductors (HTS) with transport current and/or applied field. The method in itself is a generalization of the so-called ldquoBrandt methodrdquo. In order to determine its optimal context of utilization, computation times were compared with those of the finite element method (FEM), for the case of a simple monocore superconducting tape. In order to perform an objective comparison, the same adaptive time step integration algorithm (DASPK) was used in both cases. This algorithm is built-in in the COMSOL Multiphysics package, which served as our benchmark for the FEM, whereas we had to implement it within a compiled version of the SAM based on a C proprietary code. To this end, we used the IDA solver (from the SUNDIALS package), available as a public C code. Comparisons were performed for different ldquonrdquo values for the superconducting material, and for different mesh coarseness. As a result, the SAM proved to be 10 times faster than the FEM for problems involving 300 elements in the mesh (conducting regions only), and showed equal performances with the FEM with 850-900 elements. As the number of element further grows, the SAM looses its advantage over the FEM.

Structural dynamic responses analysis applying differential quadrature method

Unconditionally stable higher-order accurate time step integration algorithms based on the differential quadrature method (DQM) for second-order initial value problems were applied and the quadrature rules of DQM, computing of the weighting coefficients and choices of sampling grid points were discussed. Some numerical examples dealing with the heat transfer problem, the second-order differential equation of imposed vibration of linear single-degree-of-freedom systems and double-degree-of-freedom systems, the nonlinear move differential equation and a beam forced by a changing load were computed, respectively. The results indicated that the algorithm can produce highly accurate solutions with minimal time consumption, and that the system total energy can remain conservative in the numerical computation.

Time step integration algorithms with predetermined coefficients for second order equations

Abstract In this paper, the effect of using the predetermined coefficients in constructing time step integration algorithms suitable for linear second order differential equations based on the weighted residual method is investigated. The second order equations are manipulated directly. The displacement approximation is assumed to be in a form of polynomial in the time domain and some of the coefficients can be predetermined from the known initial conditions. The algorithms are constructed so that the approximate solutions are equivalent to the solutions given by the transformed first order equations. If there are m predetermined coefficients (in addition to the two initial conditions) and r unknown coefficients in the displacement approximation, it is shown that the formulation is consistent with a minimum order of accuracy m + r . The maximum order of accuracy achievable is m +2 r . This can be related to the Pade approximations for the second order equations. Unconditionally stable algorithms equivalent to the generalized Pade approximations for the second order equations are presented. The order of accuracy is 2 r −1 or 2 r and it is required that m +1⩽ r . The corresponding weighting parameters, weighting functions and additional weighting parameters for the Pade and generalized Pade approximations are given explicitly.

Bi-discontinuous time step integration algorithms––Part 2: second-order equations

In this paper, bi-discontinuous time step integration algorithms for Linear second-order equations are investigated. The second-order equations are manipulated directly. Both the initial and final displacement and velocity are only weakly enforced at the beginning and at the end of the time interval . As a result, there may be discontinuous jumps for both the displacement and velocity at the two ends of the time interval. If the displacement is assumed to be a polynomial of degree n +1 within the time interval, there are n +4 unknowns in the formulation. It is found that the order of accuracy would be at least n +1 for displacement and n for velocity in general. It is shown that the relevant algorithmic parameters for the second-order equations can be related to the algorithmic parameters presented in Part 1 of this paper for the first-order equations. Unconditionally stable higher order accurate time step integration algorithms with controllable numerical dissipation equivalent to the generalized Padé approximations with an order of accuracy 2 n +3 can then be constructed systematically. The weighting parameters and the corresponding weighting functions are given explicitly. It is also shown that the accuracy of the particular solutions is compatible with the accuracy of the homogeneous solutions if the proposed weighting functions are employed.

Bi-discontinuous time step integration algorithms––Part 1: first order equations

Bi-discontinuous time step integration algorithms––Part 1: first order equations

Solving non-linear problems by complex time step methods

Recently, a new type of time step integration algorithms using complex time steps has been proposed. For linear problems, the algorithms are higher order accurate, unconditionally stable and have directly controllable numerical dissipation. Solutions with high accuracy can be generated using large time steps. In this paper, the algorithms are extended to solve non-linear problems. The pseudo-force approach is adopted in treating the non-linear terms. To maintain the solutions accuracy, the pseudo-force is reconstructed by interpolation. Special treatments are required to compute the excitation at the complex time steps. Several numerical examples are analysed. It is observed that the complex time step method can be computationally more efficient than the Newmark method when very accurate numerical solutions are required. Copyright © 2002 John Wiley & Sons, Ltd.

Solving non‐linear problems by complex time step methods

AbstractRecently, a new type of time step integration algorithms using complex time steps has been proposed. For linear problems, the algorithms are higher order accurate, unconditionally stable and have directly controllable numerical dissipation. Solutions with high accuracy can be generated using large time steps. In this paper, the algorithms are extended to solve non‐linear problems. The pseudo‐force approach is adopted in treating the non‐linear terms. To maintain the solutions accuracy, the pseudo‐force is reconstructed by interpolation. Special treatments are required to compute the excitation at the complex time steps. Several numerical examples are analysed. It is observed that the complex time step method can be computationally more efficient than the Newmark method when very accurate numerical solutions are required. Copyright © 2002 John Wiley & Sons, Ltd.

Stability and accuracy of differential quadrature method in solving dynamic problems

Abstract In this paper, the differential quadrature method is used to solve dynamic problems governed by second-order ordinary differential equations in time. The Legendre, Radau, Chebyshev, Chebyshev–Gauss–Lobatto and uniformly spaced sampling grid points are considered. Besides, two approaches using the conventional and modified differential quadrature rules to impose the initial conditions are also investigated. The stability and accuracy properties are studied by evaluating the spectral radii and truncation errors of the resultant numerical amplification matrices. It is found that higher-order accurate solutions can be obtained at the end of a time step if the Gauss and Radau sampling grid points are used. However, the conventional approach to impose the initial conditions in general only gives conditionally stable time step integration algorithms. Unconditionally stable algorithms can be obtained if the modified differential quadrature rule is used. Unfortunately, the commonly used Chebyshev–Gauss–Lobatto sampling grid points would not generate unconditionally stable algorithms.

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

AbstractNumerical 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.

UNCONDITIONALLY STABLE COLLOCATION ALGORITHMS FOR SECOND ORDER INITIAL VALUE PROBLEMS

In this paper, unconditionally stable higher order accurate time step integration algorithms suitable for second order initial value problems in collocation form are presented. The second order equations are manipulated directly. If the approximate solution is expressed as a polynomial of degree n+1, there are n unknowns to be determined after taking into account the two given initial conditions. It is well known that by suppressing the residuals of the governing equations at n distinct collocation points only, the resultant algorithms are only conditionally stable. In this paper, linear combinations of the residuals at n+1 distinct collocation points are used to solve for the n unknowns. The collocation points and the relative weights between the residuals are derived from the weighted residual method. The weighting functions are arbitrary polynomials of degree not exceeding n−1. To control the accuracy and stability properties of the resultant algorithms, the reduced integration technique is used to evaluate the integrals in the formulation. Once the reduced integration rules are decided, the equivalent collocation form can be derived. It is found that the resultant algorithms cast in the collocation form are easy to implement and can be used to tackle non-linear problems directly. Numerical examples are given to illustrate the validity of the present formulation.

Solving initial value problems by differential quadrature method?part 2: second- and higher-order equations

In Part 1 of this paper, the sampling grid points for the differential quadrature method to give unconditionally stable higher-order accurate time step integration algorithms are proposed to solve first-order initial value problems. In this paper, the differential quadrature method is extended to solve second-order initial value problems. The conventional approaches to impose the given initial conditions are discussed. A new approach to impose the given initial conditions is then presented. It is found that the proposed approach could generate unconditionally stable higher-order accurate time step integration algorithms for second-order equations directly. Furthermore, the procedure can be generalized to construct unconditionally stable higher-order accurate time step integration algorithms for third- and higher-order initial value problems without any difficulties. The computational procedures for multi-degree-of-freedom systems and non-linear problems are also discussed. As demonstrated by the numerical examples, the differential quadrature method using the proposed sampling grid points and the proposed method to impose the given initial conditions is found to be more efficient than the conventional differential quadrature method in solving initial value problems. Copyright © 2001 John Wiley & Sons, Ltd.

A multi-time step integration algorithm for structural dynamics based on the modified trapezoidal rule

An explicit multi-time step (subcycling) integration method is proposed for solving semi-discretized structural dynamics problems. Based on nodal partition theory, this method is derived from the modified trapezoidal rule method (MTM). To achieve numerical stability virtually fixed physical quantities, displacement and velocity, are assumed for the non-updated nodal groups. The energy method is used to analyze the stability of the new algorithm and the critical time step for each nodal group is determined in terms of the maximum frequency of the elements in the associated subdomain. Some example problems are evaluated to numerically examine the accuracy and stability of the algorithm.

Procedures for multi-time step integration of element-free Galerkin methods for diffusion problems

An explicit multi-time step integration algorithm is presented for the element-free Galerkin method (EFGM) for diffusion problems. To partition the problems into subdomains, two different methods are considered: a nodal partition and an integration point partition. Methods of estimating the critical time step for the one-dimensional EFGM are examined and means for estimating the stable time step for the different partitions are considered. Several numerical examples are considered to demonstrate the stability and accuracy of the methods.

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.

Weighting parameters for unconditionally stable higher-order accurate time step integration algorithms. Part 2?second-order equations

In this paper, unconditionally stable higher-order accurate time step integration algorithms suitable for linear second-order differential equations based on the weighted residual method are presented. The second-order equations are manipulated directly. As in Part 1 of this paper, instead of specifying the weighting functions, the weighting parameters are used to control the algorithm characteristics. The algorithms are at least nth-order accurate if the numerical solution for displacement is approximated by a polynomial of degree n+1 with n undetermined coefficients. By choosing the weighting parameters carefully, the order of accuracy can be improved. The generalized Padé approximations for the second-order equations are considered. The ultimate spectral radius μ is an algorithmic parameter. By relating the approximate solutions to the equivalent formulations presented in Part 1 of this paper, the required weighting parameters are found explicitly. Any set of linearly independent functions can be used to construct the corresponding weighting functions from the weighting parameters. The stabilizing weighting functions for the weighted residual method are found explicitly. To ensure higher-order accuracy in the general solution, the accuracy of the particular solution due to excitation is also examined. Copyright © 1999 John Wiley & Sons, Ltd.

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.

RESPONSES OF BLAST LOADING BY COMPLEX TIME STEP METHOD

Blast loading is often described by means of high order functions, and step-by-step time integration algorithms are commonly used to evaluate the numerical solutions. The time step size for the Newmark method has to be very small in order to integrate the high order loading accurately. Recently, a complex time step formulation has been proposed to construct unconditionally stable higher order accurate time step integration algorithms with controllable numerical dissipation where loading with high order variation can be tackled without difficulties. The responses at the end of a time step are obtained by linearly combining the responses at various complex sub-step locations with different weighting factors. In this paper, the complex time step method is extended to evaluate the responses within a time step. The required weighting factors anywhere within a time step can be worked out systematically. Besides, there are some locations within a time step with one order higher in accuracy. A procedure is also proposed to evaluate the modified excitation at various complex sub-step locations. To verify the complex time step method, a single-degree-of freedom system subject to blast loading described by a fourth order polynomial is considered in detail. A multi-degree-of-freedom system is also analyzed. Excellent performance over the Newmark method is noted. It is possible to evaluate the responses due to blast loading by using just one time step.