Time Step Integration Algorithms Research Articles

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.

Unconditionally stable higher-order accurate collocation time-step integration algorithms for first-order equations

In this paper, unconditionally stable higher-order accurate time-step integration algorithms for linear first-order differential equations based on the collocation method are presented. The amplification factor at the end of the spectrum is a controllable algorithmic parameter. The collocation parameters for unconditionally stable higher-order accurate algorithms are found to be given by the roots of a polynomial in terms of the ultimate amplification factor. In general, when the numerical solution is approximated by a polynomial of degree n, this approximation is at least nth order accurate. However, by using the above collocation parameters, the order of accuracy can be improved to 2 n−1 or 2 n. The approximate solutions are found to be equivalent to the generalized Padé approximations. Furthermore, it is shown that the accuracy of the particular solution due to excitation given by the present method is compatible with the homogeneous solutions. No modification of the collocation parameters is required.

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 Pade approximations and unconditionally stable algorithms equivalent to the generalized Pade approximations are constructed. The corresponding weighting parameters and weighting functions for the Pade and generalized Pade approximations are given explicitly. Copyright © 2000 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.

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.

MIXED TIME FINITE ELEMENTS FOR VIBRATION RESPONSE ANALYSIS

This paper presents a mixed time finite element method for vibration response analysis. The underlying variational formulation is developed from the principle of virtual work. Conventional spatial discretization techniques are adopted. Time discretization is implemented by using mixed time finite elements. The proposed formulation not only forms a unified variational basis for spatial and temporal discretization; it also derives many other robust time step integration algorithms. Unconditionally stable and high order accurate algorithms with variable numerical dissipation can be constructed systematically. Moreover, the proposed formulations can be used to solve linear as well as non-linear problems. The accuracy and stability of the derived algorithms are presented.

Simulation of lossy multiconductor transmission lines using backward Euler integration

This paper presents a fixed time-step backward Euler integration algorithm for the time-domain analysis of linear circuits containing multiconductor lossy transmission lines. The new method requires the frequency derivatives of the circuit equations at a frequency point on the positive real axis that is related to the integration time step. The transient response is obtained efficiently from a single inversion of the circuit matrix. Although it is not as accurate as trapezoidal integration, the backward Euler integration is stable and gives smooth low-pass approximations for the actual response. In this paper, it is also shown that in case of impulse function excitation, the backward Euler results are equivalent to the shifted moments of the impulse response with an appropriate scaling of the frequency-dependent variables of the circuit. Numerical examples are presented for verification of the proposed method.

N-Body Simulations of Late Stage Planetary Formation with a Simple Fragmentation Model

We present results of two-dimensional gravitationalN-body simulations of the late stage of planetary formation. This stage is characterized by the direct accretion of hundreds of lunar-sized planetesimals into planetary bodies. Our simulation code is based on the Hermite Individual Timestep integration algorithm, and gravitational interactions among all bodies are included throughout the simulations. We compare our simulation with earlier works that do not include all interactions, and we find very good agreement. A previously published collisional fragmentation model is included in our simulation to study the effects of the production of fragments on the subsequent evolution of the larger planetary bodies. It is found that for realistic two-body collisions that, according to this model, both bodies will suffer fragmentation, and that the outcome of the collision will be a relatively large core containing most of the mass and a few small fragments. We present the results of simulations that include this simple fragmentation model. They indicate that the presence of small fragments have only a small effect on the growth or orbital evolution of the large planet-sized bodies.

HIGHER ORDER TIME-STEP INTEGRATION METHODS WITH COMPLEX TIME STEPS

Unconditionally stable higher order time-step integration algorithms are presented. The algorithms are based on the Newmark method with complex time steps. The numerical results at the (complex) sub-step locations are combined linearly to give higher order accurate results at the end of the time step. The ultimate spectral radius in the high-frequency range is a controllable parameter for these algorithms. Among these algorithms, the asymptotic annihilating algorithm and the non-dissipative algorithm correspond to the first sub-diagonal and diagonal Padé approximations respectively. The characteristics of the present algorithms with various numerical dissipations are found to be in between these two algorithms. The algorithmic parameters for the third, fifth and seventh order algorithms with various numerical dissipations are given explicitly. The order of accuracy is increased by one if these algorithms are set to non-dissipative. The spectral radii, algorithm damping ratios and relative period errors are compared favourably with other higher order algorithms.

Complex-time-step Newmark methods with controllable numerical dissipation

In this paper, unconditionally stable higher-order accurate time-step integration algorithms with controllable numerical dissipation are presented. The algorithms are based on the Newmark method with complex time steps. The ultimate spectral radius (μ), the sub-step locations (βj) and the weighting factors (αj) are the algorithmic parameters. For an algorithm that is (2n−1)th order accurate, the sub-step locations which may be complex, are shown to be the roots of an nth degree polynomial. The polynomial is given explicitly in terms of n and μ. The weighting factors are then obtained by solving a system of n simultaneous equations. It is further shown that the order of accuracy is increased by one for the non-dissipative algorithms with μ=1. The stability properties of the present algorithms are studied. It is shown that if the ultimate spectral radius is set between −1 and 1, the eigenvalues of the numerical amplification matrix are complex with magnitude less than or equal to unity. The algorithms are therefore unconditionally C-stable. When the ultimate spectral radius is set to 0 or 1, the algorithms are found to be equivalent to the first sub-diagonal and diagonal Pade approximations, respectively. The present algorithms are more general as the numerical dissipation is controllable and are very suitable for parallel computers. The accuracy of the excitation responses is found to be enhanced by the present complex-time-step procedure. To maintain high-order accuracy, the excitation may need some modifications. © 1998 John Wiley & Sons, Ltd.

A Superelastic Shape-Memory-Alloy Beam Model

The present paper proposes a simple shape-memory-alloy beam model, based on the classical Euler-Bernoulli beam theory. Due to restrictive kinematical assumptions, a one-dimensional constitutive equation is used. Thus, the material behavior is described through a simple l-D phenomenological model, which allows for different material responses in tension and compression as well as for different elastic properties between austenite and martensite. Two numerical procedures are developed for the determination of the cross-section response together with a time-step integration algorithm. Then, the implementation of the shape-memory-alloy beam model within a finite-element framework is addressed. Several numerical examples are investigated to assess both the performance of the beam model and the procedure developed. The complex behavior of a typical cross section is emphasized. The results obtained from the simulation of three and four-point bending tests are compared both with experimental results and with numerical solutions obtained from a three-dimensional finite-element scheme.

Simulation of High Frequency Vehicle-Track Interactions

SUMMARY An effective time domain method for simulating the vertical vehicle-track interactions for low and high frequencies is presented. To be able to investigate the effects of local defects of the track (eg voided sleepers) on contact forces and loads and long-term deterioration of track components, a discrete track model based on finite elements is necessary. The track model developed is extremely accurate and is able to consider irregularities and nonlinearities in the track structure. The number of sleepers to be considered is chosen so that the infinite extension of the track can be dealt with approximately. For excitation due to short wavelengths like short pitch corrugation or wheel flats, nonlinear contact mechanics has to be considered. The time-step integration algorithm is based on a modal decomposition of linear substructures and is therefore absolutely stable. The method includes the effects of moving masses and is valid for train speeds up to 500 m/s. The capacity of the algorithm is demonstrated for different theoretical and practical examples.

Algorithms for molecular dynamics simulations of keV particle bombardment

A variable time step integration algorithm for carrying out molecular dynamics simulations of atomic collision cascades is proposed which evaluates the interaction forces only once per time step. The algorithm is tested on some model problems which have exact solutions and is compared against other common methods. These comparisons show that the method has good stability and accuracy. Applications to Ar+ bombardment of Cu and Si show good accuracy and improved speed to the original method [D. E. Harrison, W. L. Gay, and H. M. Effron, J. Math. Phys. 10, 1179 (1969)].

Strain Softening with Creep and Exponential Algorithm

A constitutive relation that can describe tensile strain softening with or without simultaneous creep and shrinkage is presented, and an efficient time-step numerical integration algorithm, called the exponential algorithm, is developed. Microcracking that causes strain softening is permitted to take place only within three orthogonal planes. This allows the description of strain softening by independent algebraic relations for each of three orthogonal directions, including independent unloading and reloading behavior. The strain due to strain softening is considered as additive to the strain due to creep, shrinkage and elastic deformation. The time-step formulas for numerical integration of strain softening are obtained by an exact solution of a first-order linear differential equation for stress, whose coefficients are assumed to be constant during the time step but may vary discontinuously between the steps. This algorithm is unconditionally stable and accurate even for very large time steps, and guarantees that the stress is always reduced exactly to zero as the normal tensile strain becomes very large. This algorithm, called exponential because its formulas involve exponential functions, may be combined with the well-known exponential algorithm for linear aging rate-type creep. The strain-softening model can satisfactorily represent the test data available in the literature.