# Kim, Wooram (2016-08). Improved Time Integration Algorithms for the Analysis of Structural Dynamics. Doctoral Dissertation. Thesis •
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### abstract

• In this dissertation one family of second-order and two families of higher-order time integration algorithms are newly developed. For the development of a new family of second-order time integration algorithms, the original equation of structural dynamics is rewritten as two first order differential equations and one algebraic equation. These equations are called mixed formulations, because they include three different kinds of dependent variables (i.e., the displacement, velocity, and acceleration vectors). Equal linear (for the first sub-step) and quadratic (for the second sub-step) Lagrange type interpolation functions in time are used to approximate all three variables involved in the mixed formulations, then the time finite element method and the collocation method are applied to the velocity-displacement and velocity-acceleration relations of the mixed formulations to obtain one- and two-step time integration schemes, respectively. Newly developed one- and two-step time integration schemes are combined as one complete algorithm to achieve enhanced computational features. Two collocation parameters, which are included in the complete algorithm, are optimized and restated in terms of the spectral radius in the high frequency limit (also called the ultimate spectral radius) for the practical control of algorithmic dissipation. Both linear and nonlinear numerical examples are analyzed by using the new algorithm to demonstrate enhanced performance of it. The newly developed second-order algorithm can include the Baig and Bathe method and the non-dissipative case as special cases of its family. For the development of the first family of higher-order time integration algorithms, the displacement vector is approximated over the time interval by using the Hermite interpolation functions in time. The residual vector is defined by substituting the approximated displacement vector into the equation of structural dynamics. The modified weighted residual method is applied to the residual vector. The weight parameters are used to restate the integral forms of the weighted residual statements as algebraic forms, then these parameters are optimized by using the single-degree-of- freedom problem and its exact solution to achieve improved accuracy and unconditional stability. Numerical examples are used to verify performances of the new algorithms. For the development of the second family of implicit higher-order time integration algorithms, the mixed formulations that include three time dependent variables (i.e., the displacement, velocity and acceleration vectors) are used. The equal degree Lagrange type interpolation functions in time are used to approximate the dependent variables involved in the mixed formulations, and the time finite element method and the modified weighted residual method are applied to the velocity-displacement and velocity-acceleration relations of the mixed formulations. Weight parameters are used and optimized to achieve preferable attributes of time integration algorithms. Specially considered numerical examples are used to discuss some fundamental limitations of well-known second-order accurate algorithms and to demonstrate advantages of using newly developed higher-order algorithms.
• In this dissertation one family of second-order and two families of higher-order time integration algorithms are newly developed.

For the development of a new family of second-order time integration algorithms, the original equation of structural dynamics is rewritten as two first order differential equations and one algebraic equation. These equations are called mixed formulations, because they include three different kinds of dependent variables (i.e., the displacement, velocity, and acceleration vectors). Equal linear (for the first sub-step) and quadratic (for the second sub-step) Lagrange type interpolation functions in time are used to approximate all three variables involved in the mixed formulations, then the time finite element method and the collocation method are applied to the velocity-displacement and velocity-acceleration relations of the mixed formulations to obtain one- and two-step time integration schemes, respectively. Newly developed one- and two-step time integration schemes are combined as one complete algorithm to achieve enhanced computational features. Two collocation parameters, which are included in the complete algorithm, are optimized and restated in terms of the spectral radius in the high frequency limit (also called the ultimate spectral radius) for the practical control of algorithmic dissipation. Both linear and nonlinear numerical examples are analyzed by using the new algorithm to demonstrate enhanced performance of it. The newly developed second-order algorithm can include the Baig and Bathe method and the non-dissipative case as special cases of its family.

For the development of the first family of higher-order time integration algorithms, the displacement vector is approximated over the time interval by using the Hermite interpolation functions in time. The residual vector is defined by substituting the approximated displacement vector into the equation of structural dynamics. The modified weighted residual method is applied to the residual vector. The weight parameters are used to restate the integral forms of the weighted residual statements as algebraic forms, then these parameters are optimized by using the single-degree-of- freedom problem and its exact solution to achieve improved accuracy and unconditional stability. Numerical examples are used to verify performances of the new algorithms.

For the development of the second family of implicit higher-order time integration algorithms, the mixed formulations that include three time dependent variables (i.e., the displacement, velocity and acceleration vectors) are used. The equal degree Lagrange type interpolation functions in time are used to approximate the dependent variables involved in the mixed formulations, and the time finite element method and the modified weighted residual method are applied to the velocity-displacement and velocity-acceleration relations of the mixed formulations. Weight parameters are used and optimized to achieve preferable attributes of time integration algorithms. Specially considered numerical examples are used to discuss some fundamental limitations of well-known second-order accurate algorithms and to demonstrate advantages of using newly developed higher-order algorithms.

• August 2016