Anderson Acceleration

The Anderson acceleration method, also known as the Quasi-Newton Inverse Least Squares method, is a solution stabilization method for strongly coupled FSI applications. This method can accelerate convergence and improve the stability of the solution.

The Anderson acceleration method solves the fixed-point problem for a function of the displacement at the fluid boundary. The solution of a fixed-point problem can be given by:
Figure 1. EQUATION_DISPLAY
g ( x ) = x
(4634)
Where g is a fixed-point operator which represents execution of solvers associated to fluid and solid sub-domains.
Eqn. (4634) can be written as:
Figure 2. EQUATION_DISPLAY
f ( x ) = g ( x ) − x ⁢ = 0
(4635)
Using the first-order Taylor expansion Eqn. (4635) can be written as:
f ⁢ ( x ⁢ + Δ x ) ≈ f ⁢ ( x ) + J ( x ⁢ ) Δ ⁢ x  
(4636)
where J ⁢ ( x ) is the Jacobian. The Jacobian is a square matrix of derivatives of the function f ( x ⁢ ) . The Newton step Δ ⁢ x k at each Newton iteration k is determined by:
Δ x k = − J ( x k ) − 1 ⁢ f ( x k )
(4637)
x k + 1 = x k + Δ x k
(4638)

The Jacobian is not directly available for partitioned coupled FSI. Therefore, within the quasi-Newton approach the Jacobian J k ⁢ =   J ( x k ) is approximated using a history of increments:

Figure 3. EQUATION_DISPLAY
X k = [ Δ x k − m ⁢   , ... ⁢   , Δ x k − 1 ] ∈ ℜ n × m
(4639)
Figure 4. EQUATION_DISPLAY
F k = [ Δ f k − m ⁢   , ... ⁢   , Δ f k − 1 ] ∈ ℜ n × m
(4640)

In which X k contains increments of fixed-point displacements and F k contains increments of fixed-point residuals. m corresponds to the number of increments and n is the number of unknowns at the wetted interface.

Subject to the multi-secant condition:
J k − 1 F k ⁢ = X k
(4641)
Anderson acceleration approximates the inverse Jacobian J k − 1 by minimizing:
‖ J k − 1 − J k − m − 1 ‖ F
(4642)
where J k − m − 1 = − I