Construction and application of adaptive semi-Infinite boundary element with dynamic problems on half-plane

: For the dynamic problem of half-plane, if the radiation condition at the semi-infinite boundary is not taken into account in the numerical calculation, the accuracy of the result will be affected. In this paper, the basic theory of time-domain boundary element method (TD-BEM) and the propagation characteristics of stress waves in elastic media are used to transform a semi-infinite boundary into a semi-infinite boundary element which can adjust the size of the element automatically with time-space parameters. Enabling the element to simulate radiative damping effects in the far field. At last, the efficiency of the element is verified with a half-plane example under dynamic load by comparing its results with the results of the finite element method (FEM). The verification shows that the adaptive semi-infinite element can effectively simulate the radiation conditions in the far area. And it is convenient to use TD-BEM to solve the half-plane dynamics problem.


Introduction
Most engineering problems such as structure-foundation interactions, geotechnical slopes, and tunnels involve infinite or semi-infinite domains. For the infinite domain problems, BEM can automatically satisfy the radiation condition at infinity, and only need to deal with the finite boundary. Nevertheless, for the semi-infinite domain problems, the radiation condition on the infinite boundary is difficult to satisfy and intercept parts of finite boundary for calculation, the calculation results are less accurate. Tells [1] applied the half-space Mindlin's fundamental solution to replace the traditional whole-space fundamental solution. Although it is convenient to solve the semi-infinite plane problem with regular boundary, the irregular boundary problem is not applicable. Salvadori [2] introduced the infinite boundary element on the basis of boundary discretization, used the infinite boundary element to simulate the far-field semi-infinite boundary, and described the far-field displacement through the attenuation function, which not only ensured the calculation accuracy but also greatly reduced the computational load. It is worth noting that the above studies are only based on semi-infinite elastic problems. Liang [3,4] used the indirect boundary element method based on half-space Green's function to study the wave problem of shallow buried circular tunnels in the half-space. The method used the half-space Green's function and the surface boundary conditions satisfy automatically. Only the boundary condition of cavity or tunnel is considered, the problem is simplified. However, its adaptability is poor. For the same model, when the applied dynamic load changes, the position of the virtual source surface and the number of discrete elements need to be determined again.
In order to effectively simulate the radiative damping effect of the semi-infinite boundary on the semi-infinite domain, Based on the time domain boundary element theory [5][6][7][8][9][10], combined with the propagation characteristics of stress wave in elastic medium, this essay transforms the far-field semi-infinite boundary into an adaptive semi-infinite boundary element. The element is essentially a finite element which can adjust the size of the element automatically with time-space parameters. Therefore, it can be treated numerically by the method proposed by Lei [5,6]. In order to verify the correctness of the element, it is combined with TD-BEM to solve the elastic half-plane model under dynamic load. The comparison with the results of the finite element results shows that the adaptive semi-infinite boundary element can effectively simulate the radiative damping effect in the far field. For the semi-infinite plane problem, the boundary can be divided into the finite field boundary Γ 1 , the infinite boundary   and the semi-infinite boundary Γ 2 , Γ 3 , as shown in Fig.1. The field points on the boundary   are all in the infinite distance, and the radiation conditions are automatically satisfied. Therefore, the boundary   is not needed to be considered in the calculation. Under conditions of zero initial condition and negligible body force, the time domain boundary integral equation for the elastic dynamic semi-infinite plane problem is written as:

TD-BEM formulation
Where ik c [7] is the coefficient related to the position of the source point. The displacement and traction fundamental solutions are expressed by Eq. (2) and (3), respectively, as: Where ρ is density. r is the distance between the source point P and the field point Q. c s and c d represent the secondary and the primary wave velocities, respectively. H s and H d are Heaviside functions. The , , , r n r n r r n n r Where ik δ is Kronecker function; μ and λ are Lamé constant.

Construction method
According to Fig.1, the semi-infinite boundary Γ 2 and Γ 3 are the boundary extending from one end to the infinite distance, so it is impossible to carry out conventional discrete processing. In order to reduce the number of discrete elements and ensure the calculation accuracy, this paper transforms the semi-infinite boundary Γ 2 and Γ 3 into two adaptive semi-infinite elements of space integral fields according to the propagation characteristics of the stress wave in the elastic medium (the farthest propagation distance of the stress wave at The element will change automatically with the change of the position of the source point and the time that the impulse starts the impact, which has strong adaptability and can fully consider the influence of the stress wave emitted at different times.  As the position of the source point P and the time τ that the impulse starts the impact are constantly changing, the established node a (or d) is also constantly changing. Through the position relationship between the source point and the semi-infinite element, the general coordinate equation of the node can be obtained, as shown in Fig.2. Taking the semi-infinite boundary element A as an example, the general coordinate equation of node a is: In this section, according to the propagation characteristics of stress wave in elastic medium, the semi-infinite boundary on both sides is transformed into two adaptive semi-infinite boundary elements. According to the construction conditions of semi-infinite boundary elements and the general equation of node coordinates, the corresponding semi-infinite boundary elements can be automatically established for specific time-space parameters, which is highly adaptive. The element can automatically adjust the size of the element with the stress wave propagation distance, and fully consider the influence of stress wave. Therefore, the element can simulate the far-field radiation damping effect theoretically.

Numerical dispersion
In space, according to the propagation characteristics of stress wave in elastic medium, the semi-infinite boundary on both sides is transformed into two adaptive semi-infinite boundary elements. Assuming that the traction and displacement change linearly on the semi-infinite boundary element, the interpolation function is by eq. (13): Assumed that the traction and the displacement change linearly at each time interval. For any time interval, the interpolation functions of the traction and displacement is:

Solution and assembly of influence coefficient
Through discrete processing, the semi-infinite boundary is transformed into adaptive semi-infinite boundary element. The semi-infinite boundary element is essentially a finite element that can automatically adjust the size of the element according to the time-space parameters. Therefore, the numerical calculation methods such as the calculation of the influence coefficient of the unit and the processing of the singularity are the same as those of the conventional TD-BEM [4,5], which will not be repeated here. After discretizing the time-space domain, the influence coefficients of each element are calculated, and then the influence coefficients ik g and ik h are assembled in space and time. In terms of time, the displacement and traction are discretized by linear element, the displacement influence coefficient needs to consider the influence of elements before and after the time node. At instant

Example description
In this paper, an example of semi-infinite elastic foundation under dynamic load is selected for analysis. There is a long semi-cylindrical groove along the y-axis the source point P The semi-infinite boundary element A node b The semi-infinite boundary Γ 3 ` node d The semi-infinite boundary element B node c The semi-infinite boundary Γ 2 on the elastic foundation, and a radial dynamic load is applied on the inner surface of the groove, as shown in Fig.3. The radius of the groove 0 1m r = , and the relationship between load and time is shown in Fig.4. The material parameters are:  =0.25, E=4×10 8 and ρ=3×10 3 kg/m 3 . The calculated time course is 0~0.012s, and the maximum propagation distance of the stress wave is 48m. In order to verify the correctness of the adaptive semi-infinite boundary element, the results of FEM are used for comparison. When solving with FEM, in order to prevent the influence of rebound stress wave on the displacement of the calculation node, the area within 60m from the load is taken as the calculation domain, and 44984 'PLANE42' elements are discretized. When using TD-BEM to solve, the boundary within 25m from the load center is taken as the finite boundary, and a total of 500 finite boundary elements are discretized. And an adaptive semi-infinite boundary element is added on both sides of the finite boundary to consider the radiation damping effect of the far-field boundary.
The displacement response of node at r = 20 m of the earth's surface is selected for comparison, and the results are shown in Fig.5. Due to there are many time steps and the data points are dense, in order to make the image clearer, the image is set to display only one result point every two time steps, and the abscissa time variable is dimensionless processed.
Through the comparison, the results obtained by the semi-infinite time domain boundary element method are in good agreement with those obtained by FEM. It is proved that the modified element can effectively simulate the radiative damping effect in the far field. The calculation results are shown in Fig.5.

Conclusion
In this paper, according to the propagation characteristics of stress wave in elastic medium, the semi-infinite boundary is transformed into an adaptive semi-infinite boundary element that adaptively changes with time-space parameters, and the construction method and process of the element are given. Finally, an example of elastic half-plane under dynamic load is used to verify the results. The verification shows that the results obtained by TD-BEM with the adaptive semi-infinite boundary element are in good agreement with those obtained by FEM. It is proved that the modified element can effectively simulate the radiative damping effect in the far field, and it is convenient to use TD-BEM to solve the elastic half-plane dynamic problem.