UC Berkeley Math 228B, Spring 2024, Problem Set 5
PerOlof Persson
persson@berkeley.edu
Due April 5
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In this problem set, you will study two extensions of the simple
piecewise linear Poisson solver fempoi
from the previous
problem set. First, you will extend the equations to solve the Helmholtz
equation, for simulation of wave propagation in waveguides. Next, you
will extend the Poisson solver to use quadratic elements instead of
linear.
Timeharmonic Waveguide Simulations
Consider the following 2D Helmholtz problem, for a given wave number \(k\) with normalized propagation velocity, and socalled Sommerfeld radiation conditions at the in/out boundaries: \[\begin{aligned} \nabla^2 u  k^2 u &= 0, & &\text{ in }\Omega, &&(1) \\ \boldsymbol{n} \cdot \nabla u &= 0, & &\text{ on }\Gamma_\mathrm{wall} &&(2)\\ \boldsymbol{n} \cdot \nabla u + iku &= 0, & &\text{ on }\Gamma_\mathrm{out} &&(3)\\ \boldsymbol{n} \cdot \nabla u + iku &= 2ik, & &\text{ on }\Gamma_\mathrm{in} &&(4) \end{aligned}\] Here, the domain boundary \(\Gamma = \partial \Omega\) is decomposed into the three parts \(\Gamma = \Gamma_\mathrm{wall} \cup \Gamma_\mathrm{out} \cup \Gamma_\mathrm{in}\). For a computed solution \(u\), we will also calculate its intensity at the output boundaries: \[\begin{aligned} H(u) = \int_{\Gamma_\mathrm{out}} u^2\,ds, &&(5) \end{aligned}\] where \(\cdot\) is the complex absolute value.
Derive a Galerkin finite element formulation for (1)(4), for an appropriate space \(V_h\) of continuous piecewise linear functions.
Show that the discretized system can be written in the form \(A\boldsymbol{u} = \boldsymbol{b}\), where \[\begin{aligned} A = K k^2 M + i k (B_\mathrm{in} + B_\mathrm{out}) \quad \text{and} \quad \boldsymbol{b} = 2 i k \boldsymbol{b}_\mathrm{in} &&(6) \end{aligned}\] for real matrices \(K,M,B_\mathrm{in},B_\mathrm{out}\) and a vector \(\boldsymbol{b}_\mathrm{in}\), which do not depend on the wave number \(k\). Give explicit expressions for the matrix/vector entries, involving the basis functions \(\varphi_i(\boldsymbol{x})\) for the space \(V_h\).
Show that the transmitted intensity (5) for a finite element solution \(\boldsymbol{u}\) can be calculated as \(H(\boldsymbol{u}) = \boldsymbol{u}^H B_\mathrm{out} \boldsymbol{u}\).
For the implementation, consider a model test problem with wave number \(k=6\) and a straight channel domain of dimensions \(5\times 1\): \[\begin{aligned} \Omega &= \{ 0\le x \le 5, 0\le y \le 1 \} &&(7) \\ \Gamma_\mathrm{in} &= \{x=0, 0\le y\le 1 \} &&(8)\\ \Gamma_\mathrm{out} &= \{x=5, 0\le y\le 1 \} &&(9)\\ \Gamma_\mathrm{wall} &= \{0\le x \le 5, y=0\text{ or }y=1 \} &&(10) \end{aligned}\]
Show that an exact solution to the Helmholtz problem (1)(4) for the domain (7)(10) is given by \(u_\mathrm{exact}(x,y) = e^{ikx}\).
Write a function which for a given triangular mesh
p
,t
identifies the boundary edges corresponding to the wall, the in, and the out boundaries, respectively:
ein, eout, ewall = waveguide_edges(p, t)
Use the function
all_edges()
in the Mesh utilities notebook on the course web page to find all mesh edges, then assume that the inboundary consists of all vertical edges with \(x=0\), and that the outboundary consist of all vertical edges with \(x=5\).Write a function that computes the matrices \(K,M,B_\mathrm{in},B_\mathrm{out}\) and the vector \(\boldsymbol{b}_\mathrm{in}\) for a given mesh:
K, M, Bin, Bout, bin = femhelmholtz(p, t, ein, eout)
Solve the discretized problem on the meshes generated by
pv = [0 0; 5 0; 5 1; 0 1; 0 0] p, t, e = pmesh(pv, 0.3, nref)
where
nref
ranges from \(1\) to \(4\). Compute maxnorm errors using the exact solution \(u_\mathrm{exact}\), plot errors vs. mesh size in a loglog plot, and estimate the order of convergence.
Finally, you will use your Helmholtz solver to compute a frequency response for a waveguide with two slits, see figure below. The waveguide is again of dimensions \(5\)by\(1\), and the slits are \(0.2\) units wide and \(0.8\) units deep, centered at \(x=2\) and \(x=3\).
Create a mesh for the domain using
pmesh
, withhmax
= 0.2 andnref
= 2.To look for resonance phenomena around \(k\approx 2\pi\), solve for a range of wave numbers \(k\) between \(k=6\) and \(k=6.5\), in steps of \(\Delta k = 0.01\). For each \(k\), solve the problem and calculate \(H(\boldsymbol{u})\). Plot \(H\) vs. \(k\) in a semilog plot.
Plot two of your solutions using
tplot
of the real part, for the wave numbers \(k\) corresponding to the smallest and the largest value of \(H(\boldsymbol{u})\).
Quadratic Elements for Poisson’s equation
Next you will modify the
fempoi
function to use quadratic elements instead of linear. Some details of the implementation:Assume that all triangles are straightsided
Use the following secondorder accurate integration rule for a triangle \(T_k\) with corner nodes \(\boldsymbol{x}_1,\boldsymbol{x}_2,\boldsymbol{x}_3\) and area \(A_k\): \[\begin{aligned} \int_{T_k} f(\boldsymbol{x})\,d\boldsymbol{x} \approx A_k\sum_{i=1}^3 w^g_i f(\boldsymbol{x}^g_i),\ \ % w^g_i=1/3,\ \ \boldsymbol{x}^g_i = \sum_{j=1}^3 \left(\boldsymbol{x}_j/6 + \delta_{ij}\boldsymbol{x}_j/2 \right), \ \ i=1,2,3 \end{aligned}\]
Note that
tplot
only knows how to plot linear functions, but to plot a quadratic function you can at least use it to draw linear functions between the corner nodes:tplot(p, t, u[1:size(p,1)])
First, a given mesh
p,t
has to be extended for quadratic functions by adding edge midpoints. Write a function
p2, t2, e2 = p2mesh(p, t)
which produces the arrays:
p2

: \(N\times 2\), node coordinates of original mesh nodes and the new midpoints
t2

: \(T\times 6\), localtoglobal mapping for the \(T\) triangular elements
e2

: \(E\times 1\), indices of boundary nodes, both from original mesh nodes and the midpoints
Use the function
all_edges
to find all mesh edges, and note the third outputemap
which is useful when creatingt2
.Write a function
u = fempoi2(p2, t2, e2)
which solves the Poisson problem using quadratic elements, for the mesh produced by your function
p2mesh
above.Do a convergence study as follows: Solve the problem on meshes given by
hmax = 0.3 pv = [0 0; 1 0; 1 1; 0 1; 0 0] p, t, e = pmesh(pv, hmax, nref) p2, t2, e2 = p2mesh(p, t)
with homogeneous Dirichlet conditions on all boundaries and
nref
ranging from 0 to 4. Consider the finest mesh (nref
= 4) the true solution. Calculate approximate maxnorms for all the other solutions (nref
= 0 to 3) as the maximum error at the set of nodes in the coarsest mesh (nref
= 0). Plot these errors vs. mesh size in a loglog plot, and estimate the slope.
Code Submission: Your Julia file needs to define the
functions waveguide_edges
, femhelmholtz
,
p2mesh
, fempoi2
, with exactly the requested
names and input/output arguments, as well as any other supporting
functions and variables that are required for your functions to run
correctly.