I am solving the one-dimensional wave equation over regions where the bulk modulus (and thus the wave speed) vary continuously over a region. The current version seems to assume that the material properties are constant over an FEM element. Some literature suggests that isoparametric elements can help model regions where the material properties vary continuously. Can you suggest how to model regions where the material properties vary in a continuous fashion?
In the wave equation code I am using, kappa and rho can vary across an element. The code is as follows:
eqn = 1/κ[x] D[u[t, x], {t, 2}] +
Div[-ρ[x]*Grad[u[t, x], {x}], {x}] ==
1/κ[x]*10*Exp[-50 (x^2)]*Sin[2 π f t] +
NeumannValue[0, x == 0] + NeumannValue[-Derivative[1, 0][u][t, x], x ==
xMax];
ic = {u[0,x] == 0, Derivative[1, 0][u][0, x] == 0};
NDSolve
? What is the issue? $\endgroup$