Shock and Vibration

Volume 2018, Article ID 7828267, 14 pages

https://doi.org/10.1155/2018/7828267

## Infinite Element Static-Dynamic Unified Artificial Boundary

State Key Laboratory of Eco-hydraulics in Northwest Arid Region of China, Xi’an University of Technology, Xi’an, China

Correspondence should be addressed to Zhiqiang Song; moc.621@4002qihzs

Received 23 January 2018; Accepted 21 May 2018; Published 5 July 2018

Academic Editor: Roberto Palma

Copyright © 2018 Zhiqiang Song et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The method, which obtains a static-dynamic comprehensive effect from superposing static and dynamic effects, is inapplicable to large deformation and nonlinear elastic problems under strong earthquake action. The static and dynamic effects must be analyzed in a unified way. These effects involve a static-dynamic boundary transformation problem or a static-dynamic boundary unified problem. The static-dynamic boundary conversion method is tedious. If the node restraint reaction force caused by a static boundary condition is not applied, then the model is not balanced at zero moment, and the calculation result is distorted. The static numerical solution error is large when the structure possesses tangential static force in a viscoelastic static-dynamic unified boundary. This paper proposed a new static-dynamic unified artificial boundary based on an infinite element in ABAQUS to solve static-dynamic synthesis effects conveniently and accurately. The static and dynamic mapping theories of infinite elements were introduced. The characteristic of the infinite element, which has zero displacement at faraway infinity, was discussed in theory. The equivalent nodal force calculation formula of infinite element unified boundary was deduced from an external wave input. A calculation and application program of equivalent nodal forces was developed using the Python language to complete external wave inputting. This new method does not require a static and dynamic boundary transformation and import of stress field and constraint counterforce of boundary nodes. The static calculation precision of the infinite element unified boundary is more improved than the viscoelastic static-dynamic unified boundary, especially when the static load is in the tangential direction. In addition, the foundation simulation range of finite field can be significantly reduced given the utilization of the infinite element static dynamic unified boundary. The preciseness of static calculation and dynamic calculation and static-dynamic comprehensive analysis are unaffected.

#### 1. Introduction

The simulation of a semi-infinite far-field foundation in the static and dynamic interaction of a structure-foundation system is a controversial issue in the seismic field of engineering. For static problems, considering that the elastic restoration effect of a sufficient range foundation is necessary, an artificial boundary, such as fixed or roller boundary, is typically used. For dynamic problems, a dynamic artificial boundary was used at the surface of the sufficient range foundation to simulate elastic restoration and radiation damping effects given energy dissipation in an infinitive foundation. Considerable research has been conducted on all kinds of dynamic artificial boundaries. Such types of research are mainly concentrated on two major categories, that is, local and global artificial boundaries.

The commonly used local artificial boundaries include transmission [1, 2], viscous [3], and viscoelastic boundaries [4–6]. Viscoelastic boundary, compared with viscous boundary, added a spring-damping system, which not only can dissipate extroverted waves on the boundary but also simulates an elastic restoration effect of the faraway foundation. The viscoelastic boundary has high precision and improved stability [7–9].

Infinite element is a typical global artificial boundary. In 1973, Ungless first proposed the idea of the infinite element, which is used to solve the infinite domain simulation problem [10]. Several scholars have contributed to the improvement, dissemination, and application of infinity element [11–14]. Bettess proposed mapping an infinite element called Bettess element based on a mapping between the global and the local coordinates [13]. He then summarized the research results on infinite elements and published the first monograph called Infinite Elements in 1992 [15]. In recent years, certain researchers have applied infinite element to studying the dynamic interaction of structure foundation [16–20]. Yun [21, 22] innovated the dynamic infinite element formula and studied the 2D- and 3D-layered soil-structure interaction problem in frequency and time domains. Numerous studies indicate that infinite element can easily harmonize with finite element and has remarkable advantages and practicability in simulation and approximate simulation infinite domain problems compared with boundary element and other numerical methods for solving infinite domain problems.

The effects of static and dynamic loads can be solved separately for general small deformation and linear elastic structures. The superposition of the two effects is the total effect of the structures. Superposition principle is inapplicable for large deformation and nonlinear elastic problems under strong earthquake action. Static and dynamic effects must be analyzed in a unified way. These effects involve the static-dynamic boundary transformation or static-dynamic boundary unified problem. Qi [23] improved the artificial boundary of the dynamic infinite element without considering the static-dynamic unified artificial boundary based on the infinite element. Gao described a static-dynamic boundary conversion method for large deformation and nonlinear structures [24]. First, the static effect is analyzed under static boundaries, such as fixed boundary. Second, the static boundary is replaced by the dynamic boundary, such as viscoelastic boundary. Static-stress field and boundary-node-constrained reaction obtained by static analysis are introduced. Simultaneously, the original static force loads are inputted to ensure that the structure remains in a balanced state at the dynamic calculation zero-time. Third, the dynamic load is applied to analyze a dynamic response. The static-dynamic total effect can be obtained. This method is tedious. If the node restraint reaction force caused by static boundary condition is not applied, then the model is not balanced at zero moment, and the calculation result is distorted.

Several researchers began to study the static-dynamic boundary unified problem to avoid the tedious work of the static-dynamic boundary conversion. Certain researchers apply the viscoelastic dynamic artificial boundary to the static problem directly. The viscoelastic dynamic boundary was proposed based on wave motion theory in an infinite homogeneous elastic medium. The numerical solution indicated a larger deviation than theory solution when the viscoelastic dynamic artificial boundary is used directly in a static analysis. Liu proposed a viscoelastic static-dynamic unified artificial boundary [25]. He modified the spring stiffness coefficients of the viscoelastic dynamic boundary to make the dynamic boundary suitable for static analysis. The unified boundary was used to solve the static-dynamic combination problem of a semi-infinite space body bearing normal static and point source vibration loads at the free surface. The results show that the unified boundary increases the accuracy of the static analysis. Moreover, the precision and stability of a dynamic analysis can still be ensured. However, the static numerical solution error for the tangential static force remains large. Gao studied the stress of the semi-infinite space body using the viscoelastic static-dynamic unified boundary [24]. He also observed a large deviation between the transverse stress obtained by numerical simulation and theoretical solution at the internal points in the soil. The deficiency of viscoelastic static-dynamic unified boundary will not affect the accuracy of the static analysis when the static effect is mainly caused by structural self-weight. However, the hydraulic structure, water pressure, and sand pressure flow only in a horizontal direction. This deficiency of the viscoelastic static-dynamic unified boundary makes viscoelastic static-dynamic unified boundary unsuitable for hydraulic structures.

This paper proposed a new static-dynamic unified artificial boundary based on an infinite element in ABAQUS to solve the static-dynamic synthesis effect conveniently and accurately. The characteristic of the infinite element, which has zero displacement at infinite faraway, was discussed. The equivalent nodal force calculation formula of infinite element unified boundary was deduced from the external wave input. The calculation and application program of the equivalent nodal forces was developed using the Python language to complete the external wave inputting. This new method does not require static and dynamic boundary transformation and import of stress field and constraint counterforce of boundary nodes. The static calculation precision of the infinite element unified boundary is more improved than the viscoelastic static-dynamic unified boundary, especially when the static load is in a tangential direction. In addition, the foundation simulation range of finite field can be significantly reduced given the use of infinite element static-dynamic unified boundary. The accuracies of static calculation, dynamic calculation, and static-dynamic comprehensive analysis are unaffected. The calculation efficiency is improved when infinite element static-dynamic unified boundary was used for large nonlinear analysis.

This paper is organized as follows. Section 2 introduces the static and dynamic mapping theory of infinite elements. The equivalent nodal force calculation formula of infinite element unified boundary was deduced from the external wave input. Section 3 verifies the accuracy of the infinite element unified boundary that is applied to static calculation, dynamic calculation, and static-dynamic comprehensive analysis by numerical examples of a semi-infinite body and a practical gravity dam. The results show that the new method is relatively simple and accurate. The new method is especially suitable for structures subjected to horizontal tangential static load action. The influence of foundation simulation range on the finite domain to static calculation and static-dynamic comprehensive analysis is also discussed when the infinite element static-dynamic unified boundary was used. Section 4 presents the conclusions.

#### 2. Mapping Principle and Exogenous Wave Input Method of Infinite Element

##### 2.1. Static Mapping of Infinite Element

The basic ideas of the static mapping of the infinite element are as follows. Plane or space semi-infinite domain is mapped to a finite domain by applying the mapping function in geometry. Elements in the finite domain are analyzed to calculate single stiffness matrix in accordance with finite element method. Total stiffness matrix, stress, and displacement of the structure are obtained [19]. A type of 1D mapping infinite element is introduced.

Figure 1(a) illustrates an infinite element in the 1D* X* coordinate system. Nodes 1 and 2 are in the finite domain, whereas Node 3 is in the infinite domain. The infinite element in the* X* global coordinate system is transformed into the finite element (parent element) in* ξ *local coordinate system through the mapping function, as depicted in Figure 1(b). Then, we can use finite element theory in analyzing the parent element.