MATH76109/21/2018

Having carried out preparatory work in Homework03, we consider the same problem as in Homework02, but solve it using a finite volume method, and compare solutions.

Problem.

Wave scattering by an elastic sphere submerged in an incompressible fluid is described by the equation

${\displaystyle \rho {\partial }^2𝒖/\partial t^2=(\lambda +2\mu )\nabla (\nabla \cdot 𝒖)-\mu \nabla \times (\nabla \times 𝒖)}$

(1)

where:

• $𝒖(t,𝒙)$ is the displacement [m]

• $\rho$ is the density [kg/m${}^3$]

• $\lambda ,\mu$ are the bulk, shear elastic moduli [N/m${}^2$].

Elastic media sustain two types of waves:

• longitudinal or $P$-waves (pressure waves) with wave velocity $c_p=\sqrt{(\lambda +2\mu )/\rho }$

• transverse or $S$-waves (shear waves) with wave velocity $c_s=\sqrt{\mu /\rho }$

Consider a plane pressure wave $p\left(t\right)=P\sin \left(\omega t\right)$, $$entering a cube of side $a=8$m filled with water ($\rho =1000$ kg/m${}^3$, $c_p=1500$ m/s, $c_s=0$) containing a steel sphere of radius $r=1$m ($\rho =7700$ kg/m${}^3$, $c_p=6000$ m/s, $c_s=3200$ m/s) (Fig. 1)

Figure 1. Schematic of sphere scattering problem

${\displaystyle }$

1. First consider the 2D plane geometry variant of the problem. State (1) as a system ${𝒒}_t+𝑨{𝒒}_x+𝑩{𝒒}_y=0$, with $𝒗=(v_1,v_2,0)={𝒖}_t$ the displacement velocity, plane strain conditions

   ${\displaystyle v_3=0,\frac{\partial 𝒗}{\partial x_3}=0,\frac{\partial 𝝈}{\partial x_3}=0}$

   and using the Hookean constitutive relation ${\sigma }_{ij}=\lambda u_{k,k}{\delta }_{ij}+\mu (u_{i,j}+u_{j,i})$ whose time derivative gives ${\sigma }_{ij,t}=\lambda v_{k,k}{\delta }_i+\mu (v_{i,j}+v_{j,i})$, $𝒒={\left(\begin{array}{llllll}{\sigma }_{11}& {\sigma }_{22}& {\sigma }_{33}& {\sigma }_{12}& v_1& v_2\end{array}\right)}^T$. Write $𝑨,𝑩$ and determine their eigenstructure. Solve the problem using Bearclaw.

2. Now consider the scattering problem around a cylinder, and obtain a system ${𝒒}_t+𝑨{𝒒}_x+𝑩{𝒒}_r=𝝍\left(𝒒\right)$. Repeat tasks from Question 1.

3. Finally consider the scattering problem around a sphere, obtain the system ${𝒒}_t+𝑨{𝒒}_x+𝑩{𝒒}_r=𝝌\left(𝒒\right)$. Repeat tasks from Question 1. Compare with solution from Homework02