Article Contents
Article Contents

Higher-order time-stepping schemes for fluid-structure interaction problems

• * Corresponding author: Daniele Boffi
• We consider a recently introduced formulation for fluid-structure interaction problems which makes use of a distributed Lagrange multiplier in the spirit of the fictitious domain method. In this paper we focus on time integration methods of second order based on backward differentiation formulae and on the Crank–Nicolson method. We show the stability properties of the resulting method; numerical tests confirm the theoretical results.

Mathematics Subject Classification: Primary: 65M60, 65M85; Secondary: 65M12, 74F10.

 Citation:

• Figure 1.  Geometrical configuration of the FSI problem

Figure 2.  Sparsity pattern for a matrix arising from Equation (24)

Figure 3.  Meshes for the fluid and the structure

Figure 4.  The deformed annulus 1

Figure 5.  Volume preservation over time

Figure 6.  Volume preservation over time for coarser parameters

Figure 7.  Volume preservation over time for IFEM method

Table 1.  Mesh parameters

 DOFs $\mathbf{u}_h$ DOFs $p_h$ DOFs $\mathbf{X}_h$ DOFs $\lambda_h$ coarse mesh (M = $8$) $578$ $209$ $306$ $306$ fine mesh (M = $16$) $2,178$ $801$ $1,122$ $1,122$

Table 2.  Convergence results for the fully implicit scheme on the coarse mesh

 Fluid velocity $\mathsf{BDF1}$ $\mathsf{BDF2}$ $\mathsf{CNm}$ $\mathsf{CNt}$ $\Delta t$ $L^2$ error rate $L^2$ error rate $L^2$ error rate $L^2$ error rate $0.05$ $7.63\cdot 10^{-2}$ $2.97\cdot 10^{-2}$ $2.37\cdot 10^{-1}$ $2.42\cdot 10^{-1}$ $0.025$ $4.11\cdot 10^{-2}$ $0.89$ $4.90\cdot 10^{-3}$ $2.60$ $6.24\cdot 10^{-2}$ $1.92$ $6.02\cdot 10^{-2}$ $2.00$ $0.0125$ $2.13\cdot 10^{-2}$ $0.95$ $1.13\cdot 10^{-3}$ $2.11$ $1.21\cdot 10^{-2}$ $2.36$ $1.10\cdot 10^{-2}$ $2.45$ $0.00625$ $1.08\cdot 10^{-2}$ $0.97$ $2.86\cdot 10^{-4}$ $1.98$ $2.03\cdot 10^{-3}$ $2.58$ $9.95\cdot 10^{-4}$ $3.47$ Structure deformation $\mathsf{BDF1}$ $\mathsf{BDF2}$ $\mathsf{CNm}$ $\mathsf{CNt}$ $\Delta t$ $L^2$ error rate $L^2$ error rate $L^2$ error rate $L^2$ error rate $0.05$ $1.60\cdot 10^{-3}$ $4.39\cdot 10^{-4}$ $1.43\cdot 10^{-3}$ $2.95\cdot 10^{-4}$ $0.025$ $8.40\cdot 10^{-4}$ $0.93$ $9.75\cdot 10^{-5}$ $2.17$ $7.90\cdot 10^{-4}$ $0.86$ $6.89\cdot 10^{-5}$ $2.10$ $0.0125$ $4.29\cdot 10^{-4}$ $0.97$ $2.53\cdot 10^{-5}$ $1.95$ $4.06\cdot 10^{-4}$ $0.96$ $7.53\cdot 10^{-6}$ $3.19$ $0.00625$ $2.17\cdot 10^{-4}$ $0.98$ $6.41\cdot 10^{-6}$ $1.98$ $2.04\cdot 10^{-4}$ $0.99$ $2.05\cdot 10^{-6}$ $1.88$

Table 3.  Convergence results for the fully implicit scheme on the fine mesh

 Fluid velocity $\mathsf{BDF1}$ $\mathsf{BDF2}$ $\mathsf{CNm}$ $\mathsf{CNt}$ $\Delta t$ $L^2$ error rate $L^2$ error rate $L^2$ error rate $L^2$ error rate $0.05$ $9.05\cdot 10^{-2}$ $3.62\cdot 10^{-2}$ $2.28\cdot 10^{-1}$ $2.26\cdot 10^{-1}$ $0.025$ $4.87\cdot 10^{-2}$ $0.89$ $5.05\cdot 10^{-3}$ $2.84$ $6.23\cdot 10^{-2}$ $1.87$ $6.04\cdot 10^{-2}$ $1.91$ $0.0125$ $2.54\cdot 10^{-2}$ $0.94$ $1.20\cdot 10^{-3}$ $2.07$ $2.28\cdot 10^{-2}$ $1.45$ $2.07\cdot 10^{-2}$ $1.54$ $0.00625$ $1.29\cdot 10^{-2}$ $0.98$ $3.53\cdot 10^{-4}$ $1.77$ $5.27\cdot 10^{-3}$ $2.11$ $4.03\cdot 10^{-3}$ $2.36$ Fluid velocity $\mathsf{BDF1}$ $\mathsf{BDF2}$ $\mathsf{CNm}$ $\mathsf{CNt}$ $\Delta t$ $L^2$ error rate $L^2$ error rate $L^2$ error rate $L^2$ error rate $0.05$ $1.98\cdot 10^{-3}$ $5.19\cdot 10^{-4}$ $1.65\cdot 10^{-3}$ $4.04\cdot 10^{-4}$ $0.025$ $1.05\cdot 10^{-3}$ $0.92$ $9.79\cdot 10^{-5}$ $2.41$ $9.27\cdot 10^{-4}$ $0.84$ $8.48\cdot 10^{-5}$ $2.25$ $0.0125$ $5.31\cdot 10^{-4}$ $0.99$ $3.13\cdot 10^{-5}$ $1.64$ $4.90\cdot 10^{-4}$ $0.92$ $2.47\cdot 10^{-5}$ $1.78$ $0.00625$ $2.70\cdot 10^{-4}$ $0.98$ $1.35\cdot 10^{-5}$ $1.22$ $2.50\cdot 10^{-4}$ $0.97$ $3.47\cdot 10^{-6}$ $2.83$

Table 4.  Maximum iterates of the nonlinear solver on the coarse mesh

 $\Delta t$ $\mathsf{BDF1}$ $\mathsf{BDF2}$ $\mathsf{CNm}$ $\mathsf{CNt}$ $0.05$ $5$ $5$ $4$ $7$ $0.025$ $4$ $4$ $3$ $4$ $0.0125$ $3$ $3$ $3$ $3$ $0.00625$ $3$ $3$ $2$ $3$

Table 5.  Maximum iterates of the nonlinear solver on the fine mesh

 $\Delta t$ $\mathsf{BDF1}$ $\mathsf{BDF2}$ $\mathsf{CNm}$ $\mathsf{CNt}$ $0.05$ $10$ $5$ $6$ $6$ $0.025$ $6$ $5$ $5$ $4$ $0.0125$ $6$ $4$ $4$ $4$ $0.00625$ $4$ $4$ $3$ $3$

Table 6.  Convergence results for the semi-implicit scheme on the coarse mesh

 Fluid velocity $\mathsf{BDF1}$ $\mathsf{BDF2}$ $\mathsf{CNm}$ $\mathsf{CNt}$ $\Delta t$ $L^2$ error rate $L^2$ error rate $L^2$ error rate $L^2$ error rate $0.05$ $7.78\cdot 10^{-2}$ $3.05\cdot 10^{-2}$ $2.49\cdot 10^{-1}$ $2.58\cdot 10^{-1}$ $0.025$ $4.17\cdot 10^{-2}$ $0.90$ $7.89\cdot 10^{-3}$ $1.95$ $6.24\cdot 10^{-2}$ $2.00$ $6.74\cdot 10^{-2}$ $1.94$ $0.0125$ $2.17\cdot 10^{-2}$ $0.95$ $3.14\cdot 10^{-3}$ $1.33$ $1.24\cdot 10^{-2}$ $2.33$ $2.64\cdot 10^{-2}$ $1.35$ $0.00625$ $1.10\cdot 10^{-2}$ $0.97$ $1.29\cdot 10^{-3}$ $1.29$ $2.25\cdot 10^{-3}$ $2.47$ $3.18\cdot 10^{-3}$ $3.06$ Structure deformation $\mathsf{BDF1}$ $\mathsf{BDF2}$ $\mathsf{CNm}$ $\mathsf{CNt}$ $\Delta t$ $L^2$ error rate $L^2$ error rate $L^2$ error rate $L^2$ error rate $0.05$ $1.67\cdot 10^{-3}$ $6.79\cdot 10^{-4}$ $1.44\cdot 10^{-3}$ $3.52\cdot 10^{-4}$ $0.025$ $8.65\cdot 10^{-4}$ $0.95$ $2.70\cdot 10^{-4}$ $1.33$ $7.91\cdot 10^{-4}$ $0.86$ $2.60\cdot 10^{-4}$ $0.44$ $0.0125$ $4.36\cdot 10^{-4}$ $0.99$ $1.24\cdot 10^{-4}$ $1.12$ $4.05\cdot 10^{-4}$ $0.97$ $1.53\cdot 10^{-5}$ $4.08$ $0.00625$ $2.17\cdot 10^{-4}$ $1.01$ $5.71\cdot 10^{-5}$ $1.12$ $2.05\cdot 10^{-4}$ $0.98$ $9.43\cdot 10^{-6}$ $0.70$

Table 7.  Convergence results for the semi-implicit scheme on the fine mesh

 Fluid velocity $\mathsf{BDF1}$ $\mathsf{BDF2}$ $\mathsf{CNm}$ $\mathsf{CNt}$ $\Delta t$ $L^2$ error rate $L^2$ error rate $L^2$ error rate $L^2$ error rate $0.05$ $9.18\cdot 10^{-2}$ $3.89\cdot 10^{-2}$ $2.36\cdot 10^{-1}$ $2.39\cdot 10^{-1}$ $0.025$ $5.05\cdot 10^{-2}$ $0.86$ $8.59\cdot 10^{-3}$ $2.18$ $7.54\cdot 10^{-2}$ $1.64$ $7.06\cdot 10^{-2}$ $1.76$ $0.0125$ $2.63\cdot 10^{-2}$ $0.94$ $3.32\cdot 10^{-3}$ $1.37$ $4.24\cdot 10^{-2}$ $0.83$ $2.22\cdot 10^{-2}$ $1.67$ $0.00625$ $1.33\cdot 10^{-2}$ $0.98$ $1.40\cdot 10^{-3}$ $1.24$ $2.19\cdot 10^{-2}$ $0.96$ $4.19\cdot 10^{-3}$ $2.40$ Structure deformation $\mathsf{BDF1}$ $\mathsf{BDF2}$ $\mathsf{CNm}$ $\mathsf{CNt}$ $\Delta t$ $L^2$ error rate $L^2$ error rate $L^2$ error rate $L^2$ error rate $0.05$ $2.03\cdot 10^{-3}$ $7.86\cdot 10^{-4}$ $1.81\cdot 10^{-3}$ $6.51\cdot 10^{-4}$ $0.025$ $1.06\cdot 10^{-3}$ $0.93$ $3.28\cdot 10^{-4}$ $1.26$ $9.75\cdot 10^{-4}$ $0.89$ $1.31\cdot 10^{-4}$ $2.31$ $0.0125$ $5.34\cdot 10^{-4}$ $1.00$ $1.44\cdot 10^{-4}$ $1.18$ $5.10\cdot 10^{-4}$ $0.93$ $4.82\cdot 10^{-5}$ $1.44$ $0.00625$ $2.69\cdot 10^{-4}$ $0.99$ $6.31\cdot 10^{-5}$ $1.19$ $2.55\cdot 10^{-4}$ $1.00$ $1.29\cdot 10^{-5}$ $1.90$

Table 8.  Maximum residual in the semi-implicit scheme on the coarse mesh

 $\Delta t$ $\mathsf{BDF1}$ $\mathsf{BDF2}$ $\mathsf{CNm}$ $\mathsf{CNt}$ $0.05$ $3.83\cdot 10^{-3}$ $3.83\cdot 10^{-3}$ $9.87\cdot 10^{-3}$ $9.64\cdot 10^{-2}$ $0.025$ $2.09\cdot 10^{-3}$ $2.30\cdot 10^{-3}$ $1.24\cdot 10^{-3}$ $2.17\cdot 10^{-2}$ $0.0125$ $7.41\cdot 10^{-4}$ $8.26\cdot 10^{-4}$ $3.62\cdot 10^{-4}$ $7.90\cdot 10^{-3}$ $0.00625$ $2.23\cdot 10^{-4}$ $2.45\cdot 10^{-4}$ $1.08\cdot 10^{-4}$ $9.55\cdot 10^{-4}$

Table 9.  Maximum residual in the semi-implicit scheme on the fine mesh

 $\Delta t$ $\mathsf{BDF1}$ $\mathsf{BDF2}$ $\mathsf{CNm}$ $\mathsf{CNt}$ $0.05$ $3.59\cdot 10^{-2}$ $3.59\cdot 10^{-2}$ $3.43\cdot 10^{-2}$ $3.81\cdot 10^{-2}$ $0.025$ $1.03\cdot 10^{-2}$ $1.07\cdot 10^{-2}$ $8.19\cdot 10^{-3}$ $1.02\cdot 10^{-2}$ $0.0125$ $5.28\cdot 10^{-3}$ $5.87\cdot 10^{-3}$ $1.54\cdot 10^{-3}$ $2.27\cdot 10^{-3}$ $0.00625$ $1.46\cdot 10^{-3}$ $1.44\cdot 10^{-3}$ $4.33\cdot 10^{-4}$ $5.08\cdot 10^{-4}$
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