Boundary Elements

The main problem in performing local mesh refinements relates to the need to generate a transition between refined and unrefined regions. Different approaches are available in the literature to address this issue, for example transition elements or multi−point constraints approaches [24 (link),25 (link),26 (link),27 ]. The s−refinement strategy offers the advantage of simplifying this process by allowing an element size reduction in the desired regions only. This result is achieved through the definition of an independent local/fine mesh which is superimposed to a global/coarse one [30 (link)].
Using this idea, the final FE approximation $\varphi$ can be represented as: $$\varphi =\left\{\begin{array}{c}{\varphi }_G\mathrm{in}\Omega -{\Omega }_L\hfill \\ {\varphi }_G+{\varphi }_L\mathrm{in}{\Omega }_L\hfill \end{array}\right.$$
where ${\varphi }_G$ is the global mesh solution defined in $\Omega$ , while ${\varphi }_L$ is the local mesh solution defined in ${\Omega }_L\subset \Omega$ , see Figure 5. Note that the mesh superposition technique allows for incompatible discretization between global and local mesh. This gives an extremely high level of flexibility when performing local h−refinements, as no transition regions [24 (link),25 (link),26 (link)] or multi-point constraints [27 ] are required.
This concept of solution superposition can be extended to multi−level refinements [41 (link)]. In this case, the elements of the local mesh can be further refined by superposing one over the other multiple levels of overlaid meshes. In this case, the final FE solution $\varphi$ becomes: $$\varphi =\left\{\begin{array}{c}{\varphi }_G\mathrm{in}\Omega -{\Omega }_L^{\left(1\right)}\hfill \\ {\varphi }_G+{\varphi }_L^{\left(1\right)}\mathrm{in}{\Omega }_L^{\left(1\right)}-{\Omega }_L^{\left(2\right)}\hfill \\ \dots \hfill \\ {\varphi }_G+{\varphi }_L^{\left(1\right)}+\dots +{\varphi }_L^{\left(s\right)}+\dots +{\varphi }_L^{\left(N_s\right)}\mathrm{in}{\Omega }_L^{\left(N_s\right)}\hfill \end{array}\right.$$
where ${\varphi }_L^{\left(s\right)}$ is the local solution given by the mesh at level s covering the domain ${\Omega }_L^{\left(s\right)}\subset {\Omega }_L^{(s-1)}\subset \dots \subset \Omega$ . Note that the solution on the global ${\varphi }_G$ and local meshes ${\varphi }_L^{\left(s\right)}$ can be represented by any FE scheme, such as the p−FEM presented in Section 3.1.
Two conditions, i.e., compatibility of the basis functions and their linear independency, are required to apply this multi−level decomposition of the solution field $\varphi$ .
The first condition implies $C^1$ −continuity within each elements and $C^0$ −continuity across the element boundaries. The $C^1$ −continuity is satisfied by construction. On the contrary, the inter−element continuity is not guaranteed and needs to the imposed. This is achieved by enforcing homogeneous Dirichlet boundary conditions on the boundary of the overlaid meshes, as depicted in Figure 6a.
The second condition on the linear independency is required to avoid singularities in the stiffness matrix. In general, the redundant degrees−of−freedom can be removed during the factorization process by elimination of the equations with zero pivots [30 (link)]. If the p−FEM is employed for the global and local meshes, this is avoided by ensuring that shape functions of the same type (nodal, side, face) and polynomial order p appear only once in regions with multiple meshes. This idea is graphically illustrated in Figure 6b.

The above-described mathematical model of a circular resonator with a radial exciting electric field can be supplemented with a layer of viscoelastic material of finite thickness with finite electrical conductivity, which is located on the free side of the piezodisk [25 (link)].
Now, we suppose that a plane-parallel layer of isotropic viscoelastic material with thickness f is located on the upper side of the disk. This material is characterized by two material elasticity modules c^f₁₁ and c^f₄₄, viscosity coefficients η^f₁₁ and η^f₄₄, permittivity ε^f, electrical conductivity σ^f, and density p^f. The corresponding effective tensors can be written as: $$\left[c^f\right]=\left[\begin{array}{cccc}{c}_{11}^f& c_{12}^f& c_{12}^f& 0\\ c_{12}^f& c_{11}^f& c_{12}^f& 0\\ c_{12}^f& c_{12}^f& c_{11}^f& 0\\ 0& 0& 0& c_{44}^f\end{array}\right],\left[{\eta }^f\right]=\left[\begin{array}{cccc}{\eta }_{11}^f& {\eta }_{12}^f& {\eta }_{12}^f& 0\\ {\eta }_{12}^f& {\eta }_{11}^f& {\eta }_{12}^f& 0\\ {\eta }_{12}^f& {\eta }_{12}^f& {\eta }_{11}^f& 0\\ 0& 0& 0& {\eta }_{44}^f\end{array}\right],\left[{\epsilon }^f\right]=\left[\begin{array}{cc}{\epsilon }^f-\frac{I{\sigma }^f}{\omega }& 0\\ 0& {\epsilon }^f-\frac{I{\sigma }^f}{\omega }\end{array}\right]$$
where c^f₁₂ = c^f₁₁ − 2c^f₄₄, η^f₁₂ = η^f₁₁ − 2η^f₄₄.
The distributions of the corresponding fields inside the viscoelastic layer and the piezoelectric are crosslinked using the boundary conditions of continuity: $${\begin{array}{l}{u}_i-u_i^f=0,\\ \left(T_{ij}-T_{ij}^f\right)n_j=0,\\ \phi -{\phi }^f=0,\\ \left(D_j-D_j^f\right)n_j=-\delta \end{array}|}_{z=h/2}$$
where the values with the upper index f refer to the film, n_j is the component of the boundary normal, T is the stress tensor, D is the electric induction, and δ is the density of the electric charge in the film. In this work, we assumed that δ = 0.
As shown in [25 (link)], the solution of this problem by the finite element method allows us to calculate the electrical impedance of a film-coated disk for a specified frequency ω using the known constants of the piezodisk and film. The solution of the inverse problem for this case, using the preliminarily refined material constants of the piezoceramic of the disk, allows us to determine the acoustic parameters of the material of the film covering the upper side of the disk, as well as its electrical conductivity.

The head-stem trunnion model, a conventional hip implant and its components used in this study are shown in Figure 1. In the Section 2.1 below, the hip implant models with smooth and spiral head-neck taper junctions are shown with proper diagrams. Then, the finite element model with standard boundary and loading conditions is illustrated.