## 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.

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 (

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

### 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

### 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

## 4. Statement of problem

A dam-reservoir-soil (DRS) system which is subjected to the spatially uniform horizontal (

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 ^{3}, and ^{3}, and

Figure 8 represents the accelerogram (horizontal component) recorded at a control point on the free surface; the maximum and minimum values of acceleration are

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

## 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

## References

- 1.
Hughes TJR. Analysis of transient algorithms with particular reference to stability behavior. In: Computational Methods in Mechanics. Vol. 1. Amsterdam, New York: North-Holland Publishing Co.; 1983. pp. 67-155 - 2.
Hughes TJR, Hulbert GM. Space-time finite element methods for elastodynamics: Formulations and error estimates. Computer Methods in Applied Mechanics and Engineering. 1988; 66 (3):339-363 - 3.
Li XD, Wiberg N-E. Structural dynamic analysis by a time-discontinuous galerkin finite element method. International Journal for Numerical Methods in Engineering. 1996; 39 (12):2131-2152 - 4.
Sharma V, Fujisawa K, Murakami A. Velocity-based time-discontinuous galerkin space-time finite element method for elastodynamics. Soils and Foundations. 2018; 58 (2):491-510 - 5.
Sharma V, Fujisawa K, Murakami A. Space–time fem with block-iterative algorithm for nonlinear dynamic fracture analysis of concrete gravity dam. Soil Dynamics and Earthquake Engineering. 2020; 131 :105995 - 6.
Bajer CI, Dyniewicz B. Virtual functions of the space–time finite element method in moving mass problems. Computers & Structures. 2009; 87 (7–8):444-455 - 7.
Bause M, Radu FA, Köcher U. Space–time finite element approximation of the biot poroelasticity system with iterative coupling. Computer Methods in Applied Mechanics and Engineering. 2017; 320 :745-768 - 8.
Bonelli A, Bursi OS, Mancuso M. Explicit predictor–multicorrector time discontinuous galerkin methods for non-linear dynamics. Journal of Sound and Vibration. 2002; 256 (4):695-724 - 9.
Mancuso M, Ubertini F. An efficient time discontinuous galerkin procedure for non-linear structural dynamics. Computer Methods in Applied Mechanics and Engineering. 2006; 195 (44–47):6391-6406 - 10.
Sharma V, Fujisawa K, Murakami A. Space-time finite element procedure with block-iterative algorithm for dam-reservoir-soil interaction during earthquake loading. International Journal for Numerical Methods in Engineering. 2019; 120 (3):263-282 - 11.
Hughes TJR. The Finite Element Method: Linear Static and Dynamic Finite Element Analysis. Mineola, New York: Dover Publications; 2000 - 12.
Zienkiewicz OC, Taylor RL. The Finite Element Method for Solid and Structural Mechanics. Butterworth-Heinemann; 2014 - 13.
Løkke A, Chopra AK. Direct finite element method for nonlinear analysis of semi-unbounded dam–water–foundation rock systems. Earthquake Engineering and Structural Dynamics. 2017; 46 (8):1267-1285