The second part of the analysis calculates Es*• the forces and moments caused by the seismic excitation. Roughness serves as the boundary force contribution to the interior finite element equations.
ORGANIZATION
In review, each of the proposed transmission boundary schemes has been shown to be effective for selected wave problems. Through the equilibrium equations it is shown that their underlying prescriptions are equivalent to those of the paraxial boundary.
CHAPTER 2
INTRODUCTION
ENERGY FORMS
Letting h be proportional to u in the boundary region to satisfy equation (4) is an obvious choice, but some alternatives exist. However, it does not indicate whether these boundary terms remove all energy impinging on the boundary.
THEORY OF PARAXIAL BOUNDARIES
In this case, h becomes a convective term which sends energy in the positive x-direction and eventually out of the system. In the derivation herein, the paraxial boundary idea is best introduced using the one-dimensional wave equation.
TWO-DIMENSIONAL SCALAR WAVE EQUATION
If we substitute the positive root ckx/w into equation (13), then this equation represents waves traveling in the positive-x direction. We can determine the differential equation corresponding to the first three tens of equation (17) by inspecting equation (13) and its .
BOUNDARY APPROXIMATIONS FOR TWO-DIMENSIONAL, LINEAR ELASTICITY The analysis using the elasticity equations follows along lines
The third analogy to the case of scalar waves is that now the paraxial differential equations can be derived. In fact, the term negative stiffness raises the question of whether the paraaxial equations are stable.
STABILITY ANALYSIS
For cs< cd/2 (ie, Poisson's ratio greater than 1/3), instabilities may lead to the use of the paraxial equations. The rationale for this simple solution is that the stiffness terms are the least important element of the paraxial approximation, as found in the derivation of the paraxial equations.
THEORETICAL COMPARISONS OF PARAXIAL TO VISCOUS BOUNDARIES A. The Viscous Boundary
The viscous boundary applies the ut and wt values directly to the left side of the element. Substituting the potentials in equations (48) into equations (49) yields the elastic displacements propagating in the elastic region.
ANGLE ~F INCIDENCE ANGLE ~F INCIDENCE
ANGLE ~F INCIDENCE
INCIDENCE
In fact, earlier authorsC47 •57l eliminated these amplitudes from the comparison when they multiplied them by the wave velocity time cos a (a= angle of incidence.) This new quantity measured the energy flux at the boundary; the energy propagating a 1 from the boundary was assumed to be confined there. Overall, considering all wave reflection curves, we conclude that each of the boundary schemes produces acceptable results.
ANGLE ~F INCIOENCE ANGLE ~F INCIDENCE
RAYLEIGH WAVES
Therefore, we want to determine the ability of the "body wave" paraxial equations to absorb Rayleigh wave motion. The values of the coefficients for other Poisson ratios are in the same range as those presented in Tables 1 and 2. The effectiveness of the standard viscous boundary in the transmission of Rayleigh waves is largely untested.
In a practical sense, it is not clear that using the variable coefficients will improve the absorption of Rayleigh waves. The bound causes a significant reduction in the critical time step for the explicit part of the solution algorithm (which is described in Chapter 3).
OTHER SILENT-BOUNDARY APPLICATIONS A. Spherically Symmetric Case
However, if the above considerations are relatively less important, then the boundary stresses (57) may be useful for Rayleigh wave applications. The derivation and implementation of the paraxial boundary, although not discussed here, follows from this analysis and the procedures described earlier in this chapter. If we now replace potentials (69) and (72) in equations (73) and (75), we get two equations for the solutions of two unknowns, AP and As.
For purposes of comparison with the plane strain case, we can determine the amplitudes of the potential $ used in the analysis in Sect. The vector potential,!· is related by the equation in which g x is the symbol for the convolved vector and z is the unit vector in the z direction.
ANGLE ~F
ANGLE DF INCIDENCE
ANGLE DF
RNGLE CTF INCIDENCE RNGLE CTF INCIDENCE
The analyzes of Chapter 2 suggest that the boundary schemes discussed earlier can reproduce most of the effects of an infinite domain. We discuss the problems of using a paraxial boundary and then implementing two viscous boundaries. We can evaluate the behavior of the extended-paraxial equations, for a one-dimensional mesh which is drawn in figure 1.
As ;1 moves to the left, less and less ripple is transferred to the right. This technique is one way of "weighting" the integration of finite elements in the flow direction.
VALIDATION OF NUMERICAL PROCEDURES
In all these problems and in the test examples in Chapter IV, the silent boundaries were implemented as shown in Figure 9. In the case of the viscous boundary, the black strip denotes a set of applied stresses, crxx and 'xz', which are applied to the boundary. A load is applied to the interior of the rod during the first time step and then removed.
The wave pulses are denoted by circles whose diameters represent the magnitude and direction of the particle velocities. The circles that remain at the paraxial boundary after step number 9 do so because of the attachment of the boundary nodes on.
INTERIOR REGION
All the free nodes experience some displacement as the wave passes them, but the right-hand nodes remain stationary. After checking the code with several simple, smooth analytical solutions, we test its ability to model discontinuous waves generated by a delta function load in both time and space. A byproduct of this example is that the results indicate the effectiveness of the silent-limit methods.
Miklowitz(BS) presents a derivation, based on the Cagniard-DeHoop method, of Lamb's analytical solutions and plots them along the x and z axes. We compare these results with those produced by a grid which is coarse with respect to the wavefront.
LAMB'S PA~BLEM
The comparisons between the analytical solutions and finite-element solutions using the implicit algorithm are presented in Figures 15 and 16. The analytical result shows a Rayleigh singularity where a negative infinite displacement changes instantaneously to a positive infinite displacement. When we compare the analytical and finite element solutions in the interior (x = 0.0, z = 2.75), we obtain better agreement between the methods.
The width of the initial wave pulse is still narrower than one element, but its size is finite, unlike the displacement along the surface. It leads to slightly improved accuracy inside, but adds spurious surface noise.
TIMECSECS)
TIME( SECS)
LAMB'S PACIBLEM
INTRODUCTION A. General Aim
CHAPTER 4
- DIRECT INCIDENCE OF DILATATIONAL WAVES
All the examples in this chapter were solved using the silent boundaries described in Chapter 3. Both the elastic and boundary regions were discretized with four-node elements using bilinear, isoparametric shape functions. 1This lumping of the mass matrix was done to facilitate the calculations of energy. The symmetry of the load allows the problem to be reduced to that of a conveniently resolved quarter plane.
Silent boundaries are marked with a thick black bar at the outer radial edge of the grid. This boundary absorbs almost all of the radiated energy, except for a small reflection that occurs at the inner boundary, at.
RADIAL. DILATATI~NAL PULSE
STANDARD-VISCOUS BOUNDARY, POISSON'S RATI0=.33
PULSE LOADINGS - GENERAL DISCUSSION
Time histories of the response are recorded at three points, labeled A, B or C, in each of the meshes in Figure 6. The two excitations considered are a vertical pulse that mainly generates dilational waves and a tensile pulse that generates mainly shear waves. These loads were selected for their simplicity and their relevance to the vertical and horizontal loads that occur in soil-structure interaction problems.
These are the "symmetric" boundary conditions that allow us to analyze a half space with a quarter mesh.
EXTENDED MESH
Vertical-Pulse Horizontal- Pulse
HORIZONTAL-PULSE LOADING
With these values, the dilatational waves reach the limit of the smallest grid in 9 seconds; the shear waves reach the same point in 16 seconds. Figures 8 and 9 report those displacements recorded near the lateral boundary (point B in Figure 6); The arrival of the main pulse is visible in all figures. The horizontal displacement oscillations in Figures 10 and 11 appear in all calculations, including the elongated mesh.
Each of the _broadcast limits preserves the period of these high-frequency motions, which arise from the coarseness of the mesh and the nature of the load. In displacement calculations, as shown in Figures 9 and 11, the unified (optimized) viscous boundary almost duplicates those results produced by the standard viscous boundary.
TIMEC SECS)
VERTICAL-PULSE LOAOING
Vertical load response calculations were performed without the use of numerical or material damping. Reflections from the free boundary are evident, while both quiet boundary schemes remove the outgoing waves. Vertical displacements represent the largest part of the response and are recorded at point B in Figure 23.
The same conclusion applies to the vertical displacements at the bottom of the grid, point C, as shown in Figure 24. In general, for the vertical undrained loading problem, absorbing boundaries remove most of the output energy.
VERTICAL HALF-SINE PULSE
The long-term shift observed in these figures is similar to that found in Lamb's analytical solution. The viscous and extended paraxial bounds work almost equally well for stresses, but the paraxial bound has better accuracy for vertical displacements.
FREE BOUNDARY
BOUNDARY
STEP 16
STEP 21
STEP 26
STEP 31
TIMECSECS) 30.0
DISCUSSION OF NONLINEAR WAVES
The interior of the mesh can be governed by nonlinear equations, but the region adjacent to the silent boundary must be linear. This assumption, that the governing equations are linear at the outer edges of the computational grid, is suitable for many different problems. Wave motion originating from the interaction zone and propagating to the outer boundaries can often be adequately represented by linear soil models.
However, our experience with the uniform (“optimised”) viscous limit does indicate this. that the viscous limits are relatively insensitive to the parameters a and b. The numerical results, using either the standard or the uniform viscous limit, are almost identical. We found that using the modified viscous limits, instead of the standard viscous limit, yields virtually no differences in the stresses and vertical displacements.
RAYLEIGH-WAVE EXAMPLE
The overall conclusion is that the linear boundaries can be useful in absorbing the slower moving, non-linear waves.
SMALL MESH WITH SILENT BOUNDARY
Second, the ability of the quiet boundaries to transmit Rayleigh waves must be evaluated. The horizontal and vertical displacements along the left side of the mesh are prescribed, for the entire subsequent time, according to a well-known Rayleigh-wave solution. Figures 30 and 31 illustrate most of the worst agreement between the extended-paraxial boundary and the extended mesh.
The elongated-paraxial boundary results contain spurious noise superimposed on the overall waveform, while the viscous boundary changes the period of the motions. Rayleigh mode shapes can also be used to estimate the accuracy of transmission boundaries.
RAYLEIGH WAVE
TI ME( SECS)
VER. OISP
CHAPTER 5
In the problems we studied, all silent bounds resulted in adequate to exceptional results. Tajimi, H., "Dynamic Analysis of a Structure Embedded in an Elastic Stratum", Proceedings of the Fourth World Conference on Earthquake Engineering, Santiago, Chile, 1964. Lysmer, J., "Bumped Mass Method for Rayleigh Waves", Seismological Society of America Bulletin, Vol.
Brooks, A., and Hughes, TJR, "Streamline-Upwind/Petrov-Galerkin Methods for Advection-dominated Streams", Proceedings of the Third. Mullen, R., and Belytschko, T., "Dispersion analysis of finite element semidiscretizations of the two-dimensional wave equation".
