Abstract
Finite element method (FEM) is the most extended approach for analyzing the design of the dams against earthquake motion. In such simulations, time integration schemes are employed to obtain the response of the dam at time tn+1 from the known response at time tn. To this end, it is desirable that such schemes are high-order accurate in time and remain unconditionally stable large time-step size can be employed to decrease the computation cost. Moreover, such schemes should attenuate the high-frequency components from the response of structure being studied. Keeping this in view, this chapter presents the theory of time-discontinuous space-time finite element method (ST/FEM) and its application to obtain the response of dam-reservoir system to seismic loading.
Keywords
- space-time FEM
- seismic response
- concrete dam
- time-integration
- earthquake simulation
1. Introduction
During an event of earthquake stability of dams is of paramount importance as their failure can cause immense property and environmental damages. When dam-reservoir-foundation system is subjected to the dynamic loading it causes a coupled phenomenon; ground motion and deformations in the dam generate hydrodynamic pressure in the reservoir, which, in turn, can intensify the dynamic response of the dam. Moreover, spatial-temporal variation of stresses in the dam-body depends on the dynamic interactions between the dam, reservoir, and foundation. Therefore, it becomes necessary to use numerical techniques for the safety assessment of a given dam-design against a particular ground motion.
Dynamic finite element method is the most extended approach for computing the seismic response of the dam-reservoir system to the earthquake loading [1]. In this approach finite elements are used for discretization of space domain, and basis functions are locally supported on the spatial domain of these elements and remain independent of time. Furthermore, nodal values of primary unknowns depend only on time. Accordingly, this arrangement yields a system of ordinary differential equations (ODEs) in time which is then solved by employing time-marching schemes based on the finite difference method (FDM), such as Newmark-
In dynamic finite element method (FEM), it is desirable to adopt large time-steps to decrease the computation time while solving a transient problem. Therefore, it is imperative that the time-marching scheme remains unconditionally stable and higher order accurate [1]. In addition, it should filter out the high frequency components from the response of structure. To achieve these goals, Hughes and Hulbert presented space-time finite element method (ST/FEM) for solving the elastodynamics problem [2]. In this method, displacements
However, for elastodynamics problem, ST/FEM, yields a larger system of linear equations due to due to the time-discontinuous interpolation of displacement and velocity fields. Several efforts have been made in the past to overcome this issue; both explicit [8] and implicit [9] predictor-multi-corrector iteration schemes have been proposed to solve linear and nonlinear dynamics problems. Recently, to reduce the number of unknowns in ST/FEM, a different approach is taken in which only velocity is included in primary unknowns while displacement and stresses are computed from the velocity in a post-processing step [4, 10]. To this end, the objective of the present chapter is to introduce this method (henceforth, ST/FEM) in a pedagogical manner. The rest of the chapter is organized as follows. Sections 2 and 3 deal with the fundamentals of time-discontinuous Galerkin method. Section 4 describes the dam-reservoir-soil interaction problem, and Section 5 discusses the application of ST/FEM for this problem. Lastly, Section 6 demonstrates the numerical performance of proposed method and in the last section concluding remarks are included.
2. Time-discontinuous Galerkin method (tDGM) for second order ODE
Consider a mass-spring-dashpot system as depicted in Figure 1. The governing equation of motion is described by the following second order initial value problem in time.
![](http://cdnintech.com/media/chapter/71623/1512345123/media/F1.png)
Figure 1.
Schematic diagram of the mass-spring-dashpot system.
where
In what follows, this second order ODE will be utilized to discuss the fundamental concepts behind time-discontinuous Galerkin methods (henceforth, tDGM).
2.1 Two-field tDGM
In two-field tDGM (henceforth, uv-tDGM), both displacement (
![](http://cdnintech.com/media/chapter/71623/1512345123/media/F2.png)
Figure 2.
Schematic diagram of time discontinuous approximation: (a) piecewise linear interpolation, and (b) piecewise quadratic interpolation.
where
are the discontinuous values of
The weak-form of the uv-tDGM can be stated as: find
where
Eq. (10) denotes that, in uv-tDGM, displacement-velocity compatibility relationship is satisfied in weak form.
2.2 Single field tDGM
To decrease the number of unknowns in comparison to those involved in uv-tDGM, displacement-velocity compatibility condition (cf. Eq. 6) can be explicitly satisfied and velocity can be selected as primary unknown. Henceforth, this strategy will be termed as v-tDGM. In v-tDGM,
The weak form of the v-tDGM reads: Find
Note that Eqs. (9) and (11) are identical, however, in former,
Let us now focus on the discretization of weak-form (cf. Eq. (11)) by using the locally defined piecewise linear test and trial functions,
where
Accordingly, Eq. 11 transforms into following matrix-vector form.
where
3. Numerical analysis of tDGM
In this section, numerical analysis of the tDGM schemes, (viz. uv-tDGM and v-tDGM) for the second order ODE will be performed. To assess the stability characteristics and temporal accuracy of these schemes, classical finite difference techniques will be used ([11], Chapter 9). In this context, it is sufficient to consider the following homogeneous and undamped form of Eq. (1):
3.1 Energy decay in v-tDGM
In this section it will be shown that v-tDGM is a true energy-decaying scheme. Consider Eq. (17) which represents the governing equation of a spring-mass system. The total energy (sum of kinetic and potential energy) of the system remains constant because damping and external forces are absent in the system.
Consider the time domain
Accordingly, it can be shown that
or
This shows that v-tDGM is an energy decaying time integration algorithm, in which the total energy during any time step,
To assess the energy dissipation characteristics of v-tDGM, Eq. (17) is solved with
![](http://cdnintech.com/media/chapter/71623/1512345123/media/F3.png)
Figure 3.
Energy decay characteristics of v-tDGM; (a) temporal variation of normalized total energy and (b) phase diagram obtained with different time-step sizes.
3.2 Stability characteristics of v-tDGM
In this section, to study the stability characteristics of v-tDGM, Eq. (17) is considered. The matrix-vector form corresponding to this problem is given by
where
where
To investigate the stability of v-tDGM one should look into the eigenvalues of
3.3 High-frequency response of TDG/FEM
Figure 4 plots the frequency responses of
![](http://cdnintech.com/media/chapter/71623/1512345123/media/F4.png)
Figure 4.
Frequency response of spectral radius
3.4 Accuracy of v-tDGM
In [4], it is shown that
where
Subsequently, by expanding
A direct consequence of the convergence is that the solution of Eq. (17) can be given by following expression [1]:
with
where
Further, to investigate the accuracy of v-tDGM, algorithmic damping ratio, which is a measure of amplitude decay, and relative frequency error
![](http://cdnintech.com/media/chapter/71623/1512345123/media/F5.png)
Figure 5.
Accuracy of v-tDGM: (a) algorithmic damping ratio, and (b) relative frequency error in low frequency regime (after [
4. Statement of problem
A dam-reservoir-soil (DRS) system which is subjected to the spatially uniform horizontal (
![](http://cdnintech.com/media/chapter/71623/1512345123/media/F6.png)
Figure 6.
Schematic diagram of dam-reservoir-soil (DRS) system subjected to seismic ground motion.
Further, hydrodynamic pressure distribution in the reservoir is modeled by the pressure wave equation,
with following initial and boundary conditions.
In Eq. (26),
Let us now consider the initial-boundary value problem of the solid domain which is described by,
Furthermore, following time dependent boundary conditions will be considered in Eq. (33):
In addition, solid domain is considered to be an isotropic, homogeneous, linear elastic material with
In Eqs. (31)–(34),
in which,
5. Space-time finite element method
Recently, ST/FEM is employed to solve dam-reservoir-soil interaction problem [10] in which the resultant matrix-vector form is given by
where
Further, In Eqs. (41) and (42),
where
denote mass matrix, diffusion matrix, and viscous boundary at the upstream truncated boundary, respectively. The fluid-solid coupling matrix
and right hand side spatial nodal vectors in Eqs. (41) and (42) becomes,
where
in which
the solid-fluid coupling matrix,
and right hand side spatial nodal vectors is given by,
where
Further, the force vector
Lastly, nodal values of displacement and pressure field at time
6. Numerical examples
In this section, ST/FEM with block iterative algorithm has been employed to study the response of the concrete gravity dam to the horizontal earthquake motion (see [10]). In the numerical modeling two cases are considered; (i) dam-reservoir (DR) system, in which foundation is considered to be rigid, and (ii) dam-reservoir-soil (DRS) system, in which the foundation is an elastic deformable body.
Figure 7 depicts the physical dimensions of the dam-reservoir system. Length of the reservoir in upstream direction is
![](http://cdnintech.com/media/chapter/71623/1512345123/media/F7.png)
Figure 7.
Physical dimensions of dam-reservoir system (after [
Figure 8 represents the accelerogram (horizontal component) recorded at a control point on the free surface; the maximum and minimum values of acceleration are
![](http://cdnintech.com/media/chapter/71623/1512345123/media/F8.png)
Figure 8.
Time history of horizontal component of ground motion recorded at free-surface.
![](http://cdnintech.com/media/chapter/71623/1512345123/media/F9.png)
Figure 9.
Acceleration response at the crest of dam in DR and DRS system; time history of (a) horizontal component and (b) vertical component of acceleration, and Fourier spectrum of (c) horizontal component and (d) vertical component of acceleration (after [
Interestingly, in both cases, it is observed that the critical location for pressure is at the base of the dam. Figure 10 presents the evolution of
![](http://cdnintech.com/media/chapter/71623/1512345123/media/F10.png)
Figure 10.
Temporal response of (a) normalized hydrodynamic pressure, (b) principal tensile stress and (c) principal compressive stress at the base of dam in DR and DRS system (after [
![](http://cdnintech.com/media/chapter/71623/1512345123/media/F11.png)
Figure 11.
(a) and (c): Hydrodynamic pressure field in the reservoir, and (b) and (d): magnified deformed configuration of dam at times t = 18:02 s and t = 18:08 s (after [
7. Conclusions
In this chapter, novel concepts of time-discontinuous Galerkin (tDGM) method is presented. A method called v-tDGM is derived to solve second order ODEs in time. In this method velocity is the primary unknown and it remain discontinuous at discrete times. Thereby, the time-continuity of velocity is satisfied in a weak sense. However, displacement is obtained by time-integration of the velocity in a post-processing step by virtue of which it is continuous in time. It is demonstrated that the present method is unconditionally stable and third-order accurate in time for linear interpolation of velocity in time. Therefore, it can be stated that the numerical characteristics of the v-ST/FEM scheme, therefore, make it highly suitable for computing the response of bodies subjected to dynamic loading conditions, such as fast-moving loads, impulsive loading, and long-duration seismic loading, among others.
Subsequently, ST/FEM is used to compute the response of a dam-reservoir-soil (DRS) system to the earthquake loading while considering all types of dynamic interactions. An auxiliary variable
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