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\section{3D Wave Propagation} 
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\label{WAVE CHAP} 
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In this next example we want to calculate the displacement field $u\hackscore{i}$ for any time $t>0$ by solving the wave equation: 
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\index{wave equation} 
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\begin{eqnarray}\label{WAVE general problem} 
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\rho u\hackscore{i,tt}  \sigma\hackscore{ij,j}=0 
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\end{eqnarray} 
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in a three dimensional block of length $L$ in $x\hackscore{0}$ 
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and $x\hackscore{1}$ direction and height $H$ 
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in $x\hackscore{2}$ direction. $\rho$ is the known density which may be a function of its location. 
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$\sigma\hackscore{ij}$ is the stress field \index{stress} which in case of an isotropic, linear elastic material is given by 
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\begin{eqnarray} \label{WAVE stress} 
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\sigma\hackscore{ij} & = & \lambda u\hackscore{k,k} \delta\hackscore{ij} + \mu ( u\hackscore{i,j} + u\hackscore{j,i}) 
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\end{eqnarray} 
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where $\lambda$ and $\mu$ are the Lame coefficients 
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\index{Lame coefficients} and $\delta\hackscore{ij}$ denotes the Kronecker symbol\index{Kronecker symbol}. 
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On the boundary the normal stress is given by 
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\begin{eqnarray} \label{WAVE natural} 
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\sigma\hackscore{ij}n\hackscore{j}=0 
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\end{eqnarray} 
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for all time $t>0$. 
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\begin{figure}[t!] 
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\centerline{\includegraphics[angle=90,width=4.in]{figures/waveu}} 
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\caption{Displacement at Source Point} 
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\label{WAVE FIG 1.2} 
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\end{figure} 
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\begin{figure}[t!] 
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\centerline{\includegraphics[angle=90,width=4.in]{figures/wavea}} 
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\caption{Acceleration at Source Point} 
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\label{WAVE FIG 1.1} 
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\end{figure} 
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Here we are modelling a point source at the point $x\hackscore C$ in the $x\hackscore{0}$direction 
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which raise from zero to a maximum displacement $U\hackscore 0$ with in $\alpha$ seconds and then falls back to zero over the same period. In mathematical terms we use 
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\begin{eqnarray} \label{WAVE source} 
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u\hackscore 0(x\hackscore C,t)= U\hackscore 0 \frac{t^2}{\alpha^2} e^{1\frac{t^2}{\alpha^2}} \ 
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\end{eqnarray} 
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for all $t\ge0$. In the simulations we will choose $\alpha=0.3$, see Figure~\ref{WAVE FIG 1.2} 
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and will apply the source as a constraint in a in a sphere of small radius around the point 
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$x\hackscore C$. 
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We use an explicit time integration scheme to calculate the displacement field $u$ at 
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certain time marks $t^{(n)}$ where $t^{(n)}=t^{(n1)}+h$ with time step size $h>0$. In the following the upper index ${(n)}$ refers to values at time $t^{(n)}$. We use the Verlet scheme \index{Verlet scheme} with constant time step size $h$ 
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which is defined by 
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\begin{eqnarray} \label{WAVE dyn 2} 
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u^{(n)}=2u^{(n1)}u^{(n2)} + h^2 a^{(n)} \\ 
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\end{eqnarray} 
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for all $n=2,3,\ldots$. It is designed to solve a system of equations of the form 
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\begin{eqnarray} \label{WAVE dyn 2b} 
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u\hackscore{,tt}=G(u) 
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\end{eqnarray} 
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where one sets $a^{(n)}=G(u^{(n1)})$. 
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In our case $a^{(n)}$ is given by 
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\begin{eqnarray}\label{WAVE dyn 3} 
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\rho a^{(n)}\hackscore{i}=\sigma^{(n1)}\hackscore{ij,j} 
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\end{eqnarray} 
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and boundary conditions 
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\begin{eqnarray} \label{WAVE natural at n} 
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\sigma^{(n1)}\hackscore{ij}n\hackscore{j}=0 
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\end{eqnarray} 
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derived from \eqn{WAVE natural} where 
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\begin{eqnarray} \label{WAVE dyn 3a} 
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\sigma\hackscore{ij}^{(n1)} & = & \lambda u^{(n1)}\hackscore{k,k} \delta\hackscore{ij} + \mu ( u^{(n1)}\hackscore{i,j} + u^{(n1)}\hackscore{j,i}). 
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\end{eqnarray} 
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We also need to apply the constraint 
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\begin{eqnarray} \label{WAVE dyn 3aa} 
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a^{(n)}\hackscore{0}(x\hackscore C,t)= U\hackscore{0} 
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\frac{2}{\alpha^2} \left( 1 5 \frac{t^2}{\alpha^2} +2 \frac{t^4}{\alpha^4} 
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\right) e^{1\frac{t^2}{\alpha^2}} 
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\end{eqnarray} 
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which is derived from equation~\ref{WAVE source} by calculating the second order time derivative, 
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see~\ref{WAVE FIG 1.2}. Now we have converted our problem for displacement, $u^{(n)}$, into a problem for 
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acceleration, $a^{(n)}$, which now depends 
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on the solution at the previous two time steps, $u^{(n1)}$ and $u^{(n2)}$. 
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In each time step we have to solve this problem to get the acceleration $a^{(n)}$, and we will 
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use the \LinearPDE class of the \linearPDEs to do so. The general form of the PDE defined through 
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the \LinearPDE class is discussed in \Sec{SEC LinearPDE}. The form which is relevant here is 
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\begin{equation}\label{WAVE dyn 100} 
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D\hackscore{ij} a^{(n)}\hackscore{j} =  X\hackscore{ij,j}\; . 
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\end{equation} 
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The natural boundary condition 
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\begin{equation}\label{WAVE dyn 101} 
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n\hackscore{j}X\hackscore{ij} =0 
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\end{equation} 
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is used. 
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With $u=a^{(n)}$ we can identify the values to be assigned to $D$ and $X$: 
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\begin{equation}\label{WAVE dyn 6} 
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\begin{array}{ll} 
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D\hackscore{ij}=\rho \delta\hackscore{ij}& 
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X\hackscore{ij}=\sigma^{(n1)}\hackscore{ij} \\ 
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\end{array} 
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\end{equation} 
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Moreover we need to define the location $r$ where the constraint~\ref{WAVE dyn 3aa} is applied. We will apply 
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the constraint on a small sphere of radius $R$ around $x\hackscore C$ (we will use 3p.c. of the width of the domain): 
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\begin{equation}\label{WAVE dyn 6 1} 
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r\hackscore{i}(x) = 
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\left\{ 
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\begin{array}{rc} 
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1 & \xx_c\\le R \\ 
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0 & \mbox{otherwise} 
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\end{array} 
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\right. 
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\end{equation} 
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The following script defines a the function \function{wavePropagation} which 
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implements the Verlet scheme to solve our wave propagation problem. 
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The argument \var{domain} which is a \Domain class object 
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defines the domain of the problem. \var{h} and \var{tend} are the time step size 
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and the end time of the simulation. \var{lam}, \var{mu} and 
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\var{rho} are material properties. 
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\begin{python} 
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def wavePropagation(domain,h,tend,lam,mu,rho,U0): 
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x=domain.getX() 
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# ... open new PDE ... 
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mypde=LinearPDE(domain) 
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mypde.getSolverOptions().setSolverMethod(mypde.getSolverOptions().LUMPING) 
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kronecker=identity(mypde.getDim()) 
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# spherical source at middle of bottom face 
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xc=[width/2.,width/2.,0.] 
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# define small radius around point xc 
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# Lsup(x) returns the maximum value of the argument x 
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src_radius = 0.03*width 
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print "src_radius = ",src_radius 
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dunit=numpy.array([1.,0.,0.]) # defines direction of point source 
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mypde.setValue(D=kronecker*rho, q=whereNegative(length(xxc)src_radius)*dunit) 
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# ... set initial values .... 
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n=0 
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# initial value of displacement at point source is constant (U0=0.01) 
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# for first two time steps 
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u=Vector(0.,Solution(domain)) 
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u_last=Vector(0.,Solution(domain)) 
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t=0 
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# define the location of the point source 
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L=Locator(domain,xc) 
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# find potential at point source 
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u_pc=L.getValue(u) 
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print "u at point charge=",u_pc 
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# open file to save displacement at point source 
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u_pc_data=FileWriter('./data/U_pc.out') 
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u_pc_data.write("%f %f %f %f\n"%(t,u_pc[0],u_pc[1],u_pc[2])) 
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while t<tend: 
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t+=h 
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# ... get current stress .... 
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g=grad(u) 
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stress=lam*trace(g)*kronecker+mu*(g+transpose(g)) 
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# ... get new acceleration .... 
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amplitude=U0*2*exp(1)/alpha**2*(15*(t/alpha)**2+2*(t/alpha)**4)*exp((t/alpha)**2) 
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mypde.setValue(X=stress, r=dunit*amplitude) 
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a=mypde.getSolution() 
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# ... get new displacement ... 
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u_new=2*uu_last+h**2*a 
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# ... shift displacements .... 
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u_last=u 
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u=u_new 
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n+=1 
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print n,"th time step t ",t 
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u_pc=L.getValue(u) 
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print "u at point charge=",u_pc 
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# save displacements at point source to file for t > 0 
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u_pc_data.write("%f %f %f %f\n"%(t,u_pc[0],u_pc[1],u_pc[2])) 
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# ... save current acceleration in units of gravity and displacements 
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if n==1 or n%10==0: saveVTK("./data/usoln.%i.vtu"%(n/10),acceleration=length(a)/9.81, 
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displacement = length(u), tensor = stress, Ux = u[0] ) 
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u_pc_data.close() 
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\end{python} 
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Notice that 
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all coefficients of the PDE which are independent of time $t$ are set outside the \code{while} 
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loop. This is very efficient since it allows the \LinearPDE class to reuse information as much as possible 
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when iterating over time. 
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The statement 
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\begin{python} 
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mypde.getSolverOptions().setSolverMethod(mypde.getSolverOptions().LUMPING) 
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\end{python} 
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switches on the use of an aggressive approximation of the PDE operator as a diagonal matrix 
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formed from the coefficient \var{D}. 
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The approximation allows, at the cost of 
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additional error, very fast 
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solution of the PDE. When using lumping the presence of \var{A}, \var{B} or \var{C} will produce wrong results. 
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There are a few new \escript functions in this example: 
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\code{grad(u)} returns the gradient $u\hackscore{i,j}$ of $u$ (in fact \var{grad(g)[i,j]}=$=u\hackscore{i,j}$). 
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There are restrictions on the argument of the \function{grad} function, for instance 
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the statement \code{grad(grad(u))} will raise an exception. 
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\code{trace(g)} returns the sum of the main diagonal elements \var{g[k,k]} of \var{g} 
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and \code{transpose(g)} returns the matrix transpose of \var{g} (ie. $\var{transpose(g)[i,j]}=\var{g[j,i]}$). 
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We initialize the values of \code{u} and \code{u_last} to be zero. It is important 
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to initialize both with the \SolutionFS \FunctionSpace as they have to be seen as solutions of PDEs from previous time steps. In fact, the \function{grad} does not accept arguments with a certain \FunctionSpace, for more details see \Sec{SEC ESCRIPT DATA}. 
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The \class{Locator} is designed to extract values at a given location (in this case $x\hackscore C$) from functions such as the displacement vector \code{u}. Typically the \class{Locator} is used in the following form: 
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\begin{python} 
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L=Locator(domain,xc) 
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u=... 
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u_pc=L.getValue(u) 
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\end{python} 
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The return value \code{u_pc} is the value of \code{u} at the location \code{xc}\footnote{In fact the finite element node which is closest to the given position. The usage of \class{Locator} is MPI save.}. 
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The output \code{U_pc.out} and \code{vtu} files are saved in a directory called \code{data}. 
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The \code{U_pc.out} stores 4 columns of data: $t,u\hackscore x,u\hackscore y,u\hackscore z$ 
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respectively, where $u\hackscore x,u\hackscore y,u\hackscore z$ are the $x\hackscore 0,x\hackscore 1,x\hackscore 2$ components of 
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the displacement vector $u$ at the point source. These can be 
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plotted easily using any plotting package. In gnuplot the command: 
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\begin{verbatim} 
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plot 'U_pc.out' u 1:2 title 'U_x' w l lw 2, 'U_pc.out' u 1:3 title 'U_y' w l lw 2, 
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'U_pc.out' u 1:4 title 'U_z' w l lw 2 
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\end{verbatim} 
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will reproduce Figure~\ref{WAVE FIG 1}. It is pointed out that we are not using the 
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standart \PYTHON \function{open} to create the file \code{U_pc.out} as it is not 
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when running \escript under MPI, see chapter~\ref{EXECUTION} for more details. 
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One of the big advantages of the Verlet scheme is the fact that the problem to be solved 
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in each time step is very simple and does not involve any spatial derivatives (which is what allows us to use lumping in this simulation). 
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The problem becomes so simple because we use the stress from the last time step rather then the stress which is 
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actually present at the current time step. Schemes using this approach are called an explicit time integration 
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schemes \index{explicit scheme} \index{time integration!explicit}. The 
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backward Euler scheme we have used in \Chap{DIFFUSION CHAP} is 
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an example of an implicit scheme 
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\index{implicit scheme} \index{time integration!implicit}. In this case one uses the actual status of 
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each variable at a particular time rather then values from previous time steps. This will lead to a problem which is 
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more expensive to solve, in particular for nonlinear problems. 
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Although 
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explicit time integration schemes are cheap to finalize a single time step, they need significantly smaller time 
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steps then implicit schemes and can suffer from stability problems. Therefore they need a 
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very careful selection of the time step size $h$. 
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An easy, heuristic way of choosing an appropriate 
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time step size is the Courant condition \index{Courant condition} \index{explicit scheme!Courant condition} 
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which says that within a time step a information should not travel further than a cell used in the 
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discretization scheme. In the case of the wave equation the velocity of a (p) wave is given as 
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$\sqrt{\frac{\lambda+2\mu}{\rho}}$ so one should choose $h$ from 
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\begin{eqnarray}\label{WAVE dyn 66} 
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h= \frac{1}{5} \sqrt{\frac{\rho}{\lambda+2\mu}} \Delta x 
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\end{eqnarray} 
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where $\Delta x$ is the cell diameter. The factor $\frac{1}{5}$ is a safety factor considering the heuristics of 
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the formula. 
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The following script uses the \code{wavePropagation} function to solve the 
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wave equation for a point source located at the bottom face of a block. The width of the block in 
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each direction on the bottom face is $10\mbox{km}$ ($x\hackscore 0$ and $x\hackscore 1$ directions ie. \code{l0} and \code{l1}). 
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The \code{ne} gives the number of elements in $x\hackscore{0}$ and $x\hackscore 1$ directions. 
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The depth of the block is aligned with the $x\hackscore{2}$direction. 
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The depth (\code{l2}) of the block in 
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the $x\hackscore{2}$direction is chosen so that there are 10 elements and the magnitude of the 
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of the depth is chosen such that the elements 
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become cubic. We chose 10 for the number of elements in $x\hackscore{2}$direction so that the 
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computation would be faster. \code{Brick($n\hackscore 0, n\hackscore 1, n\hackscore 2,l\hackscore 0,l\hackscore 1,l\hackscore 2$)} is an \finley function which creates a rectangular mesh 
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with $n\hackscore 0 \times n\hackscore 1 \times n\hackscore2$ elements over the brick $[0,l\hackscore 0] \times [0,l\hackscore 1] \times [0,l\hackscore 2]$. 
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\begin{python} 
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from esys.finley import Brick 
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ne=32 # number of cells in x_0 and x_1 directions 
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width=10000. # length in x_0 and x_1 directions 
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lam=3.462e9 
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mu=3.462e9 
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rho=1154. 
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tend=60 
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U0=0.01 # amplitude of point source 
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mydomain=Brick(ne,ne,10,l0=width,l1=width,l2=10.*width/32.) 
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h=(1./5.)*inf(sqrt(rho/(lam+2*mu))*inf(domain.getSize()) 
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print "time step size = ",h 
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wavePropagation(mydomain,h,tend,lam,mu,rho,U0) 
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\end{python} 
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The \function{domain.getSize()} return the local element size $\Delta x$. The 
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\function{inf} makes sure that the Courant condition~\ref{WAVE dyn 66} olds everywhere in the domain. 
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The script is available as \file{wave.py} in the \ExampleDirectory \index{scripts!\file{wave.py}}. Before running the code make sure you have created a directory called \code{data} in the current 
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working directory. 
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To visualize the results from the data directory: 
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\begin{verbatim} mayavi d usoln.1.vtu m SurfaceMap &\end{verbatim} You can rotate this figure by clicking on it with the mouse and moving it around. 
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Again use \code{Configure Data} to move backwards and forward in time, and 
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also to choose the results (acceleration, displacement or $u\hackscore x$) by using \code{Select Scalar}. Figure \ref{WAVE FIG 2} shows the results for the displacement at various time steps. 
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\begin{figure}[t!] 
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\centerline{\includegraphics[width=4.in]{figures/WavePC}} 
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\caption{Amplitude at Point source} 
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\label{WAVE FIG 1} 
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\end{figure} 
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\begin{figure}[t] 
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\begin{center} 
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\includegraphics[width=2in]{figures/Wavet0} 
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\includegraphics[width=2in]{figures/Wavet1} 
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\includegraphics[width=2in]{figures/Wavet3} 
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\includegraphics[width=2in]{figures/Wavet10} 
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\includegraphics[width=2in]{figures/Wavet30} 
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\includegraphics[width=2in]{figures/Wavet288} 
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\end{center} 
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\caption{Selected time steps ($n = 0,1,30,100,300$ and $2880$) of a wave propagation over a $10\mbox{km} \times 10\mbox{km} \times 3.125\mbox{km}$ block 
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from a point source initially at $(5\mbox{km},5\mbox{km},0)$ 
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with time step size $h=0.02083$. Color represents the displacement. 
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Here the view is oriented onto the bottom face. 
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\label{WAVE FIG 2}} 
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\end{figure} 