TIME INTEGRATION ALGORITHMS
Objective
In our study we wish to eliminate some limitations of existing higher-order algorithms. To design time integration algorithms with preferable these features, two kinds of modified time finite elements have been considered. To this end, we propose a time finite element procedure based on (a) the Hermite interpolation functions in time, (b) the residual defined from original second-order structural dynamics equation, (c) the modified weighted residual method and (d) the weight parameters, to overcome some limitations of existing weighted residual method based algorithms. We also propose another time finite element procedure based on (a) the Lagrange interpolation functions in time, (b) two residuals defined from unconventionally rewritten first-order equations, (c) the modified weighted residual method and (d) the weight parameters to overcome all limitations of the differential quadrature method proposed by Fung.
Background
In general, an algorithm of 3rd- or higher-order accuracy is called the higher-order accurate algorithm. Over the past four decades, many higher-order accurate time integration algorithms have been developed based on various numerical methods to overcome limitations of second-order accurate algorithms. The Newmark approximation based sub-stepping methods, the variational method, the weighted method, the collocation method, and the differential quadrature method have been used for the development of higher-order accurate time integration algorithms.
Various higer-order accurate algorithms and proposed methods
Technical Approach
In our first study we approximate the displacement vector by using the Hermite type of interpolation functions and directly manipulate the equation of structural dynamics to define the residual vector. To minimize the number of weight parameters in our algorithms, we use two different order of time derivatives of the residual vector in the modified weighted residual statements. The equation of structural dynamics and time derivatives of it evaluated at the time nodes are also used to elimination the second- and higher-order nodal time derivatives included in the Hermite approximation of the displacement. Through this unique set of computational frame of work, we expect our newly developed algorithms to be more efficient and intuitive than recently developed higher-order accurate algorithms.
A schematic representation of time element obtained from the pth-order Hermite interpolation functions
In the second study, we present a systematic and unified procedure which can be used for the development of general (2n-1)th and (2n)th-order accurate algorithms, n being the number of unknown displacement vectors included. To this end, the modified weighted residual method which minimizes the residual vectors and the time derivatives of them is considered. We note that the equation of structural dynamics is not used in the residual minimization procedure. Instead of directly using the equation of structural dynamics, in defining the residual vector, we rewrite Eq. (1.1) into a set of lower order equations by introducing two additional variables (i.e., the velocity and acceleration vectors). Then newly obtained velocity-displacement and acceleration-velocity relations are directly manipulated to find discrete relations between included variables (i.e., the displacement, velocity and acceleration vectors). In our case, the unique computational structure which is similar to the differential quadrature method is achieved through the direct manipulation of the velocity-displacement and acceleration-velocity relations, and unconditional stability and controllable algorithmic dissipation are realized through the optimization of weight parameters which are used to rewrite integral forms of the weighted residual statements into algebraic forms. Through the optimization, all weight parameters are stated in terms of one free parameter which can be used for the specification of algorithmic dissipation level. Since we use the same degree of Lagrange interpolation functions (both equally spaced and specially spaced) for the approximation of participating variables, determination of sampling points and reconstruction of the solutions at the end of the time interval is not required.
A schematic representation of the time element obtained from the 4th-degree specially spaced Lagrange interpolation functions
Reuslts and Discussion
Comparison of high frequency filtering capability of various algorithms
Advantages of using higher-order accurate alogorithms in long-term analysis
Comparison of numerical solutions of highly nonlinear simple pendulum. Second-order algorithms failed to describe correct movement of the pendulum in highly nonlinear case.
Comparison of numerical solutions of highly nonlinear simple pendulum. Only inaccurate solutions are obtained even with very small size of time step in second-order accurate algorithms.
The new algorithms developed by using the first proposed procedure can provide (a) the unconditional stability, (b) the controllable algorithmic dissipation, (c) easy computer implementation, and (d) efficient equation solving. Application of current algorithms to nonlinear cases is possible, but it is not practical as discussed in the text due to the direct manipulation of the equation of linear structural dynamics. This is due to the direct manipulation of the semi-discrete structural dynamics equations in the weighted residual statements. Other existing weighted residual based algorithms cannot be fully extended to nonlinear cases either, because the linear structural dynamic equation has been also manipulated in the traditional weighted residual statements.
Algorithms obtained from the second proposed procedure can be used like algorithms of the differential quadrature method for time integration of the structural dynamics equations. And new algorithms can provide the same computational performance of the equivalent algorithms of the Fung’s method without any additional computations (computation of quadrature points and reconstruction of solution at the end of the time step) as required in the Fung’s method. In fact, algorithms can be written down in a ready-to-use form, and it can be readily implemented into computer code. Since the computational structures of new algorithms are similar to those of the differential quadrature method, it can be easily extended to nonlinear problems without any modification. And the first order or any general order equations in time also can be tackled without difficulty by using current algorithms.
References
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[3] H. M. Hilber. Analysis and design of numerical integration methods in structural dynamics. PhD thesis, University of California Berkely, 1976.
[4] T. J. R. Hughes. The finite element method: linear static and dynamic finite element analysis. Courier Corporation, 2012.
[5] G. M. Hulbert. A unified set of single-step asymptotic annihilation algorithms for structural dynamics. Computer Methods in Applied Mechanics and Engineering,
113(1):1-9, 1994.
[6] T. C. Fung. Solving initial value problems by differential quadrature method part 2: second-and higher-order equations. International Journal for Numerical Methods in Engineering, 50(6):1429{1454, 2001.
[7] T. C. Fung. Weighting parameters for unconditionally stable higher-order accurate time step integration algorithms. part 2-second-order equations. International journal for numerical methods in engineering, 45(8):971-1006, 1999.
[8] T. C. Fung. On the equivalence of the time domain differential quadrature method and the dissipative Runge Kutta collocation method. International journal for numerical methods in engineering, 53(2):409-431, 2002.