Surface-based acoustic-structural medium interaction

Abaqus provides two alternatives for modeling interaction between acoustic and structural media: surface-based interaction or acoustic interface (ASIn) elements. Both are available in Abaqus/Standard, but only the surface-based capability is available in Abaqus/Explicit. If the special-purpose interface elements (ASIn) are used, interacting structural and acoustic nodes must be shared by the two meshes. The surface-based capability can be used for structural and acoustic meshes that have different node numbering and whose surface meshes may not be spatially coincident. The ease of use and low computational cost of the surface-based procedure make it preferable to the element-based approach.

This page discusses:

The equations on the contact interface between the structure and the acoustic medium
Solid/structural main, fluid secondary
Fluid main, solid/structural secondary

The equations on the contact interface between the structure and the acoustic medium

In the surface-based method the tractions and volumetric acceleration fluxes are computed between structural and acoustic media. In place of consistent distributed tractions or fluxes on both media, one side (identified as the “secondary”) receives point tractions/fluxes based on interpolation with the shape functions from the other (“main”) side. Either the acoustic fluid or the structural solid can be the secondary or main, and no Lagrange multipliers are introduced in the formulation. The basis for deciding which to make secondary or main is discussed in Acoustic, Shock, and Coupled Acoustic-Structural Analysis.

The transient expressions for the coupled acoustic-structural problem are

\begin{matrix} \int_{V_{f}} [δ p (\frac{1}{K_{f}} \ddot{p} + \frac{r}{ρ_{f} K_{f}} \dot{p}) + & \frac{1}{ρ_{f}} \frac{\partial δ p}{\partial x} \cdot \frac{\partial p}{\partial x}] d V + \int ​_{S_{fs} \cup S_{frs}} δ p n ​^{-} \cdot \frac{\partial p}{\partial x} d S - \int ​_{S_{ft}} δ p a ​_{i n} d S \\ + \int_{S_{fr} \cup S_{frs}} δ p (\frac{r}{ρ_{f}} \frac{1}{c_{1}} p + (\frac{r}{ρ_{f}} \frac{1}{k_{1}} + \frac{1}{c_{1}}) \dot{p} + \frac{1}{k_{1}} \ddot{p}) d S = 0 \end{matrix}

(acoustic medium) and

\begin{matrix} \int_{V} δ ε : σ d V + \int_{V} α_{c} ρ δ u^{m} \cdot {\dot{u}}^{m} d V & + \int_{V} ρ δ u^{m} \cdot {\ddot{u}}^{m} d V \\ + \int_{S_{fs} \cup S_{frs}} δ u^{m} \cdot n^{-} p d S & - \int_{S_{t}} δ u^{m} \cdot t d S = 0 \end{matrix}

(structural medium), where $n^{-}$ is the normal vector pointing into the fluid. The fluid-solid surface consists of the union of the directly coupled fluid-solid region, $S_{fs}$ , and a region coupled via a “reactive” acoustic surface or impedance boundary, $S_{frs}$ . Of primary interest here are those terms integrated over $S_{fs} \cup S_{frs}$ , which couple the two variational equations. The fluid impedance integral, over $S_{fr} \cup S_{frs}$ , depends only on the acoustic pressure field and its variations, so it is unaffected by the contact with the solid. The derivation for the steady-state case is formally identical to the transient case and will not be discussed here. For details of the differences in transient and steady-state acoustics in Abaqus, see Coupled acoustic-structural medium analysis.

When ASIn elements are used (see Acoustic Interface Elements), the formulation requires that the fluid and solid elements be geometrically and nodally conformal so that the shape functions for the structural displacements and the acoustic pressures are identical. The shape functions are integrated using standard methods to yield element matrices of dimension equal to the number of surface nodes on the element. The complete fluid-solid coupling matrices are formed by the sum over the element faces; that is, a standard element assembly operation. The two final coupling matrices have the sparsity pattern of the coupled fluid-solid element faces.

In surface-based coupling the interaction surface $S_{fs} \cup S_{frs}$ is formed by the boundary between possibly nonconforming structural and acoustic meshes. Therefore, the fluid-solid coupling matrix cannot be broken up into a sum over element faces as simply as in the ASIn case. To derive the coupling matrices in the surface-based procedure, we use a variation of the main-secondary procedure used in small-sliding contact (see Small-sliding interaction between bodies). At the start of an analysis, the projections $x_{N}$ of secondary nodes onto the main surface are found, and the areas $A_{N}$ and normals $n^{-} (x_{N})$ associated with the secondary nodes are computed. The projections are points $p (x_{N})$ on the main surface; main nodes in the vicinity of this projection are identified. Variables at the secondary nodes $x_{N}$ are then interpolated from variables at the identified main surface nodes near the projection $p (x_{N})$ .

Since the physical degrees of freedom for the fluid and solid meshes are different, two cases must be treated. The two cases handle the discretization of the coupling terms differently.

Solid/structural main, fluid secondary

If the fluid medium surface is designated as the secondary surface, we constrain values at each fluid node to be an average of the values at nearby main surface nodes (see Figure 1). The pointwise fluid-solid coupling condition,

\frac{1}{ρ_{f}} \frac{\partial p}{\partial x} \cdot n^{-} + {\ddot{u}}^{m} \cdot n^{-} = 0,

is enforced at the secondary nodes, resulting in displacement degrees of freedom added to the fluid secondary surface. These secondary displacements are constrained by the main displacements and thereby eliminated; the secondary pressures are not constrained directly.

Hence, the fluid equation coupling term

\int_{S_{fs} \cup S_{frs}} δ p n^{-} \cdot \frac{\partial p}{\partial x} d S

is equal to

- \int_{S_{fs} \cup S_{frs}} δ p n^{-} \cdot {\ddot{u}}^{m} d S .

This term is now approximated, at the secondary node level, by the interpolated values of structural displacements at the nearby main nodes times the area of the secondary node:

\int_{S_{fs} \cup S_{frs}} δ p n^{-} \cdot {\ddot{u}}^{m} d S \approx A_{N} [\sum_{i} n^{-} (x_{N}) \cdot N^{i} (p (x_{N})) {\ddot{u}}_{i}^{m}],

where $N^{i} (p (x_{N}))$ is the main surface interpolant evaluated at the projection of the secondary node, ${\ddot{u}}_{i}^{m}$ are the structural accelerations at the main nodes, and $n^{-} (x_{N})$ is the normal vector, pointing into the fluid, evaluated at the secondary node. The summation extends over all main nodes $i$ in the vicinity of the secondary node projection $p (x_{N})$ (see Figure 1). The computation is repeated for each secondary node $x_{N}$ on the surface $S_{fs} \cup S_{frs}$ and assembled to form the entire coupling matrix.

Similarly, the contribution to the pressure coupling term in the structural equation due to secondary node $x_{N}$ is approximated by

\int_{S_{fs} \cup S_{frs}} δ u^{m} \cdot n^{-} p d S \approx p_{N} A_{N} [\sum_{i} n^{-} (x_{N}) \cdot N^{i} (p (x_{N}))],

where $p_{N}$ is the acoustic pressure at secondary node $x_{N}$ , and the sum is again over the main nodes $i$ in the vicinity of the secondary node projection.

These expressions for the coupling terms result in matrices that are the transpose of each other. The normal vectors at the secondary surface are used, so these vectors must be well-defined (see About Surfaces).

Fluid main, solid/structural secondary

If the solid medium is designated as the secondary, the values on this surface are constrained to equal values interpolated from the main surface. The pointwise fluid-solid coupling condition is again enforced at the secondary nodes, resulting in acoustic pressure degrees of freedom added to the solid secondary surface (see Figure 2). These secondary pressures are constrained by the main surface acoustic pressures and eliminated; the secondary displacements are not constrained directly. Hence, the contribution to the coupling term in the acoustic equation for a single secondary node $x_{N}$ is approximated by

\int_{S_{fs} \cup S_{frs}} δ p n^{-} \cdot {\ddot{u}}^{m} d S \approx [A_{N} n^{-} (x_{N}) \cdot {\ddot{u}}_{N}^{m}] [\sum_{i} H^{i} (p (x_{N}))],

where ${\ddot{u}}_{N}^{m}$ is the structural acceleration at the secondary node and $H^{i} (p (x_{N}))$ are the interpolants on the fluid (main) surface evaluated at the projection $p (x_{N})$ . The summation extends over the main nodes $i$ in the vicinity of the secondary node projection and is repeated for all solid/structural nodes in the surface $S_{fs} \cup S_{frs}$ to compute the entire coupling matrix.

The contribution of a secondary node to the coupling term in the structural equation is approximated by

\int_{S_{fs} \cup S_{frs}} δ u^{m} \cdot n^{-} p d S \approx A_{N} \sum_{i} H^{i} (p (x_{N})) p_{i},

where $p_{i}$ is the pressure at main node $i$ and the sum is again over the main nodes in the vicinity of the secondary node projection $p (x_{N})$ .