Issue 30

L. Bing et alii, Frattura ed Integrità Strutturale, 30 (2014) 526-536; DOI: 10.3221/IGF-ESIS.30.63 527 T HE EXPLICIT FINITE ELEMENT SOLUTION OF THE SEISMIC WAVE FIELD henzhen Children’s Palace, which is under design is located at the foot of Lotus Hill. Its seismic response is mainly affected by the seismic wave propagation, which is a question of near field wave. Solution of seismic wave field in the area of engineering is the key to the earthquake disaster process simulation of the Children’s Palace. For three- dimensional simulation for complex engineering, finite element method is generally used to realize the integral solution of wave equation. The solution ways of finite elements can be divided into two kinds of implicit and explicit. The Explicit Central Difference Method Based on system kinematics basic theorem, wave equation can be expressed as differential equation of motion of Incremental Lagrange format [11] as follows:    int  ext damp Ma f f f (1) In the equation,  a is the acceleration of mesh nodes; int , ext f f are internal forces and external forces of mesh nodes respectively; damp f is the damping force; M is the mass matrix. The lumped mass matrix form is adopted in the program and is defined as follows:         T M ρ dV (2) In the equation,    is the matrix of  i ,  i selects 1 within the area of i and 0 outside the area. In the program, for hexahedral 8 node unit, the volume of node region of i is 1 8 of its neighboring units. For ease of matrix inversion, local damping force is adopted, defined damp f as follows:   int  damp ext f α f f signa (3) In the equation,  signa is sign of grid node's speed and direction.  is the local damping coefficient,    D . D is the critical damping ratio. For geotechnical materials, the scope of D is generally 2%~5%. For the explicit solution of Eq. (1) there are many methods. And central difference method with second order accuracy is adopted in this paper, as shown in Fig. 1: 0  t 1  n t 2/1  n t n t  n t 2/1  n t 1  n t 2/1   n t 2/1   n t Figure 1 : Timeline of central difference method Assume that displacement, velocity and acceleration of 1 2 0,  n t ,t , ,t are known, then, displacement, velocity and acceleration of  1 n t can be calculated by central difference approximation, as follows: S