Direct differentiation of the quasi-incompressible fluid formulation of fluid–structure interaction using the PFEM

Abstract

Accurate and efficient response sensitivities for fluid–structure interaction (FSI) simulations are important for assessing the uncertain response of coastal and off-shore structures to hydrodynamic loading. To compute gradients efficiently via the direct differentiation method (DDM) for the fully incompressible fluid formulation, approximations of the sensitivity equations are necessary, leading to inaccuracies of the computed gradients when the geometry of the fluid mesh changes rapidly between successive time steps or the fluid viscosity is nonzero. To maintain accuracy of the sensitivity computations, a quasi-incompressible fluid is assumed for the response analysis of FSI using the particle finite element method and DDM is applied to this formulation, resulting in linearized equations for the response sensitivity that are consistent with those used to compute the response. Both the response and the response sensitivity can be solved using the same unified fractional step method. FSI simulations show that although the response using the quasi-incompressible and incompressible fluid formulations is similar, only the quasi-incompressible approach gives accurate response sensitivity for viscous, turbulent flows regardless of time step size.

Appendices

Appendix 1: MINI element

The MINI fluid element adds a bubble node for velocity at its center in order to satisfy the LBB condition [1].

Fig. 9

Pressure and velocity nodes of the MINI fluid element

Using the nodes in Fig. 9, the shape functions for velocity and pressure in a 2D MINI element are defined as

$$\begin{aligned} N_1^\mathrm{v} =&L_1,\quad N_2^\mathrm{v}=L_2,\quad N_3^\mathrm{v}=L_3,\quad N_4^\mathrm{v}=27L_1L_2L_3 \end{aligned}$$

(48)

$$\begin{aligned} N_1^\mathrm{p} =&L_1,\quad N_2^\mathrm{p}=L_2,\quad N_3^\mathrm{p}=L_3 \end{aligned}$$

(49)

where $L_1$, $L_2$, and $L_3$ are area coordinates [34]

$$\begin{aligned} L_1+L_2+L_3=1 \end{aligned}$$

(50)

The spatial derivatives of the velocity and pressure shape functions used in the viscous matrix and gradient operator defined in Eqs. (14) and (17) are evaluated as

$$\begin{aligned} \frac{\partial {N^\mathrm{v}_a}}{\partial {x_i}}=\frac{\partial {N^\mathrm{p}_a}}{\partial {x_i}}=\frac{1}{A} \begin{bmatrix} c_a\\d_a \end{bmatrix},\quad a=1,2,3 \end{aligned}$$

(51)

and

$$\begin{aligned} \frac{\partial {N_\mathrm{v}^4}}{\partial {x_i}}=\frac{27}{A} \begin{bmatrix} c_1L_2L_3+c_2L_3L_1+c_3L_1L_2\\ d_1L_2L_3+d_2L_3L_1+d_3L_1L_2 \end{bmatrix} \end{aligned}$$

(52)

where A is twice the element area

$$\begin{aligned} A = x_2y_3-x_3y_2+x_3y_1-x_1y_3+x_1y_2-x_2y_1 \end{aligned}$$

(53)

and the intermediate variables are

$$\begin{aligned} \begin{aligned}&c_1 = y_2-y_3,\quad c_2=y_3-y_1,\quad c_3=y_1-y_2\\&d_1 = x_3-x_2,\quad d_2=x_1-x_3,\quad d_3=x_2-x_1 \end{aligned} \end{aligned}$$

(54)

The current coordinates $x_i$ and $y_i$ of each corner node are determined from the nodal displacements relative to their initial coordinates, $x_i^0$ and $y_i^0$, at the start of the simulation

$$\begin{aligned} \begin{aligned} x_1&= x^0_1+u_1,\quad x_2 = x^0_2+u_2,\quad x_3 = x^0_3+u_3\\ y_1&= y^0_1+w_1,\quad y_2 = y^0_2+w_2,\quad y_3 = y^0_3+w_3 \end{aligned} \end{aligned}$$

(55)

where $u_i$ and $w_i$ are the horizontal and vertical nodal displacements. Using the shape functions of the MINI element, the mass matrix in Eq. (13) can be expressed in closed-form as

$$\begin{aligned} {\mathbf M}_{ab}&= \frac{\rho A}{12}{\mathbf I}_2, \quad a=b,\qquad {\mathbf M}_{ab} = \frac{\rho A}{24}{\mathbf I}_2, \quad a \ne b \end{aligned}$$

(56)

$$\begin{aligned} {\mathbf M}_{4b}&= {\mathbf M}_{a4} = \frac{3\rho A}{40}{\mathbf I}_2,\qquad {\mathbf M}_{44} = \frac{81\rho A}{560}{\mathbf I}_2, \end{aligned}$$

(57)

while the fluid viscous matrix in Eq. (14) is

$$\begin{aligned} {\mathbf K}_{{ab}}&=\frac{\mu }{6A} \begin{bmatrix} 4c_ac_b+3d_ad_b&3d_ac_b-2c_ad_b\\ 3c_ad_b-2d_ac_b&4d_ad_b+3c_ac_b \end{bmatrix},\qquad \nonumber \\&a,b=1,2,3 \end{aligned}$$

(58)

$$\begin{aligned} {\mathbf K}_{{4b}}&={\mathbf K}_{{a4}}={\mathbf 0} \end{aligned}$$

(59)

$$\begin{aligned} {\mathbf K}_{{44}}&=\frac{27\mu }{40A} \begin{bmatrix} 4\sum (c_a)^2+3\sum (d_a)^2&\sum (c_ad_a)\\ \sum (c_ad_a)&4\sum (d_a)^2+3\sum (c_a)^2 \end{bmatrix} \end{aligned}$$

(60)

The gradient operator in Eq. (17) is expressed as

$$\begin{aligned} {\mathbf G}_{ab} = \frac{1}{6} \begin{bmatrix} c_a\\d_a \end{bmatrix},\quad {\mathbf G}_{4b} = -\frac{9}{40} \begin{bmatrix} c_b\\d_b \end{bmatrix},\qquad a,b=1,2,3 \end{aligned}$$

(61)

and the external force vector in Eq. (19) is

$$\begin{aligned} {\mathbf F}^\mathrm{ext}_a= \frac{\rho A}{6}{\mathbf b},\quad {\mathbf F}^\mathrm{ext}_4=\frac{9\rho A}{40}{\mathbf b},\qquad a=1,2,3 \end{aligned}$$

(62)

Finally, the pressure mass matrix in Eq. (20) is evaluated and lumped

$$\begin{aligned} {\mathbf M}_\mathrm{p}=\frac{A}{6\kappa } \begin{bmatrix} 1&\quad 0&\quad 0\\ 0&\quad 1&\quad 0\\ 0&\quad 0&\quad 1 \end{bmatrix} \end{aligned}$$

(63)

The expressions in Eqs. (56)–(63) will be used to show how the sensitivity equations are obtained from differentiation of the response equations.

Appendix 2: Evaluation of geometric derivatives

The geometric derivatives in Eqs. (33) and (34) are evaluated exactly using the MINI element formulation. This appendix demonstrates evaluation of the derivatives in matrices ${\mathbf H}$ and ${\mathbf T}$ but can be extended to any other fluid elements. First, the derivatives of the element area A defined in Eq. (53) are evaluated

$$\begin{aligned} \frac{\partial {A}}{\partial {u_1}}= & {} c_1,\quad \frac{\partial {A}}{\partial {u_2}} = c_2,\quad \frac{\partial {A}}{\partial {u_3}} = c_3 \end{aligned}$$

(64)

$$\begin{aligned} \frac{\partial {A}}{\partial {w_1}}= & {} d_1,\quad \frac{\partial {A}}{\partial {w_2}} = d_2,\quad \frac{\partial {A}}{\partial {w_3}} = d_3 \end{aligned}$$

(65)

Then, the derivatives of the variables defined in Eq. (54) are evaluated, for example,

$$\begin{aligned} \frac{\partial {c_1}}{\partial {u_1}}&= 0,\quad \frac{\partial {c_1}}{\partial {u_2}} = 0,\quad \frac{\partial {c_1}}{\partial {u_3}} = 0 \end{aligned}$$

(66)

$$\begin{aligned} \frac{\partial {c_1}}{\partial {w_1}}&= 0,\quad \frac{\partial {c_1}}{\partial {w_2}} = 1,\quad \frac{\partial {c_1}}{\partial {w_3}} = -1 \end{aligned}$$

(67)

The derivatives of $c_2$, $c_3$, $d_1$, $d_2$, and $d_3$ can be obtained similarly. With the derivatives in Eqs. (64 65)–(67), the matrices ${\mathbf H}$ and ${\mathbf T}$ in Eqs. (35) and (36) can be evaluated on a column by column basis. For the 2D MINI element, both ${\mathbf H}$ and ${\mathbf T}$ have six columns, each of which corresponds to the evaluation of the partial derivatives in Eqs. (35) and (36) with respect to $u_1$, $u_2$, $u_3$, $w_1$, $w_2$, and $w_3$ for columns one through six, respectively. For instance, the first column of the matrices ${\mathbf H}$ and ${\mathbf T}$ corresponds to the displacement $u_1$

$$\begin{aligned} {\mathbf H}_{1}&= \frac{\partial {\mathbf M}}{\partial {u_1}}\dot{\mathbf v}+ \frac{\partial {\mathbf K}}{\partial {u_1}}{\mathbf v}- \frac{\partial {\mathbf G}}{\partial {u_1}}{\mathbf p}- \frac{\partial {{\mathbf F}^\mathrm{ext}}}{\partial {u_1}} \end{aligned}$$

(68)

$$\begin{aligned} {\mathbf T}_{1}&= \frac{\partial {{\mathbf M}_\mathrm{p}}}{\partial {u_1}}\dot{\mathbf p}+ \frac{\partial {{\mathbf G}^\mathrm{T}}}{\partial {u_1}}{\mathbf v} \end{aligned}$$

(69)

where

$$\begin{aligned} \frac{\partial {\mathbf M}}{\partial {u_1}}= & {} \frac{\partial {\mathbf M}}{\partial {A}}\frac{\partial {A}}{\partial {u_1}},\quad \nonumber \\ \frac{\partial {{\mathbf F}^\mathrm{ext}}}{\partial {u_1}}= & {} \frac{\partial {{\mathbf F}^\mathrm{ext}}}{\partial {A}}\frac{\partial {A}}{\partial {u_1}},\quad \frac{\partial {{\mathbf M}_\mathrm{p}}}{\partial {u_1}} = \frac{\partial {{\mathbf M}_\mathrm{p}}}{\partial {A}}\frac{\partial {A}}{\partial {u_1}}\quad \end{aligned}$$

(70)

$$\begin{aligned} \frac{\partial {\mathbf K}}{\partial {u_1}}= & {} \frac{\partial {\mathbf K}}{\partial {A}}\frac{\partial {A}}{\partial {u_1}}+ \sum _{a=1}^3\left( \frac{\partial {\mathbf K}}{\partial {c_a}}\frac{\partial {c_a}}{\partial {u_1}} \right) \nonumber \\&+\sum _{a=1}^3\left( \frac{\partial {\mathbf K}}{\partial {d_a}}\frac{\partial {d_a}}{\partial {u_1}} \right) \end{aligned}$$

(71)

$$\begin{aligned} \frac{\partial {{\mathbf G}}}{\partial {u_1}}= & {} \sum _{a=1}^3\left( \frac{\partial {{\mathbf G}}}{\partial {c_a}}\frac{\partial {c_a}}{\partial {u_1}} \right) + \sum _{a=1}^3\left( \frac{\partial {{\mathbf G}}}{\partial {d_a}}\frac{\partial {d_a}}{\partial {u_1}} \right) \end{aligned}$$

(72)

$$\begin{aligned} \frac{\partial {{\mathbf G}^{T}}}{\partial {u_1}}= & {} \sum _{a=1}^3\left( \frac{\partial {{\mathbf G}^{T}}}{\partial {c_a}}\frac{\partial {c_a}}{\partial {u_1}} \right) + \sum _{a=1}^3\left( \frac{\partial {{\mathbf G}^{T}}}{\partial {d_a}}\frac{\partial {d_a}}{\partial {u_1}} \right) \end{aligned}$$

(73)

The derivatives $\frac{\partial {\mathbf M}}{\partial {A}}$, $\frac{\partial {\mathbf K}}{\partial {A}}$, $\frac{\partial {\mathbf K}}{\partial {c_a}}$, etc. are evaluated based on Eqs. (56)–(63).