Modeling Methods for Silent Boundaries in Infinite MediaMike Ross26 February 2004ASEN 5519-006: Fluid-Structure InteractionAerospace Engineering Sciences- University of Colorado at Boulder1 IntroductionIn several problems of interest in computational modeling there is a media that is either infinite orsemi-infinite. Some examples include: modeling the ocean when doing ship analysis, modelingthe soil underneath a footing, and modeling the air around an airplane. However, every finiteelement model must be terminated at some finite boundary. Unfortunately, for problems involvingwave propagation analysis, the usual finite boundary of the finite element model will cause theelastic waves to be reflected and superimposed with the progressing waves. Therefore, we wishto create a boundary, which is perfectly radiating to outgoing waves (no spurious reflections) andtransparent to incoming waves. In addition, we would like the boundary to be as close to the finitestructure as possible for computational efficiency.A simple solution to the problem is to move the boundary a great distance away from thefinite structure so that the boundary does not influence the results. However, this violates theconcept of computational efficiency. Hence, we need an artificial boundary condition to simulatea model without any finite boundary. Through a literature research there appears to be two ba-sic approaches to this boundary condition. The first is to create boundary elements with specialproperties to absorb the outgoing waves. The second approach is to create infinite elements. Thisreport will discuss three concepts of the boundary element approach: plane wave approximation,viscous damping boundary method, and perfectly matched layers. A brief introduction into infiniteelements will be presented as well.2 Plane Wave ApproximationThe plane wave approximation (PWA) is a boundary element method for creating a silent bound-ary. It is the lowest order member of the early-time approximation family. An early-time ap-proximation approaches exactness as t → 0 [1]. It is valid for short acoustic wavelengths (highfrequencies) and is useful for modeling early stages of the transient interaction [3]. Derivations ofthe early-time approximations can be found in the paper by Felippa [1]. These early-time approxi-mations do not treat added mass affects, which are valid for long acoustic wave lengths. To includethe added mass affects one would want to use a doubly asymptotic approximation (DAA) [4]. Theadvantage of using an early-time approximation instead of the DAA is that it is less expensive interms of memory storage and easier to implement.1Figure 1: PWA used on the far left of the fluidImplementation of the PWA is fairly straightforward. It is a boundary element that is con-structed on the finite extent of the infinite domain. For simplicity, it is best to construct the PWAelements so that the nodes coincide with the nodes of the media being modeled. An example of theuse of the PWA is shown in figure (1). Here a dam and the associated fluid are modeled. Instead ofmodeling the fluid to the shoreline on the opposite end of the dam, we have used a PWA on the leftside of the fluid. The system will experience an earthquake in the longitudinal direction. In orderto evaluate the dam’s response accurately, we do not want reflecting waves from the boundary.Once again, for this model the PWA elements lie on the left most face of the brick elements of thefluid, and the PWA nodes coincide with the fluid nodes.In order to prevent reflecting waves the PWA node velocities are modeled asp = ρcv,where p is a vector of pressures corresponding to the degrees of freedom (DOF), ρ is the densityof the media, c is the speed of sound in the media, and v are the DOF velocity components. Thus,the velocity of the PWA nodes are determined by the pressure acting at the nodes. From the aboveformulation we can easily see that the PWA acts a damper to the incoming pressure; thus, reducingthe reflection of the pressure wave back towards the structure. Hence, interpretation in terms ofmechanical components of the PWA fluid element is nothing more than a ρc dashpot [1].Thus,for modeling the interaction between the infinite media and the PWA, the infinite media sendspressures to the PWA and the PWA in turn send velocity (or displacement) values of the nodesback to the media.For the example of the dam-fluid system under seismic excitation, pressure waves were exam-ined as they encountered the PWA. During the video of this phenomenon one could easily see thatthe pressure waves were in fact absorbed by the PWA.2Figure 2: Primary WaveFigure 3: Secondary WaveIn summary, the plane wave approximation is an ideal silent boundary for plane waves prop-agating through a fluid media. Its implementation in terms of boundary elements is simple andstraight forward. It works well for a staggered solution procedure. It is computationally inexpen-sive in comparison to the DAA silent boundary method. However, it does not treat added massaffects.3 Viscous Damping Boundary MethodIn the previous section we discussed a silent boundary that is useful for fluid media. Next, weexamine a similar method, known as the viscous damping boundary method (VDB), that is usefulfor elastic infinite media. The VDB is similar to the plane wave approximation method in thatthe VDB uses the concept of applying viscous dampers to the DOF on the boundary element.However, for an elastic media, such as soil, there is primary waves and secondary waves thattravel through the media. Thus, the speed of these waves is used instead of the speed of sound inthe fluid, which was used for the PWA. Primary waves are compressional waves that travel throughsolids, liquids, and gases. Secondary waves are shear waves that only travel through solids (seefigure 2 and figure 3).In order to develop a boundary condition that ensures that all energy arriving at the boundary isabsorbed, Lysmer and Kuhlemeyer [6] found that the most promising expression for the boundarycondition is expressed by the following equations.σzz= aρVp˙uzPrimary waveτzx= bρVs˙uxSecondary waveτyz= bρVs˙uySecondary waveThese equations are formulated for an incident wave consisting of a primary and secondary wavethat act at an angle θ from the z-axis as shown in figure (4) [2]. In the above equations, ρ is thedensity of the media, ˙ux, ˙uy, ˙uzrepresent the velocities in the x,y,z-direction, Vpand Vsare the3velocities of