Comparison of Different Quasi-Newton Techniques for Coupling of Black Box Solvers

Abstract

Fluid-structure interaction (FSI) problems are frequently solved using partitioned simulation techniques with black-box solvers, reusing reliable and optimized codes. These problems can principally be reduced to solving a root-finding problem. In case of strong coupling, pure Gauss-Seidel iterations between the structure and flow solvers are unstable for lower modes. In these cases, quasi-Newton techniques are used, which construct an approximation of the Jacobian or its inverse by reusing information from previous iterations and time steps. Four different quasi-Newton techniques are compared: the interface quasi-Newton algorithm with an approximation for the inverse of the Jacobian from a least-squares model (IQN-ILS), the interface block quasi-Newton algorithm with approximate Jacobians from least-squares models (IBQN-LS), the interface quasi-Newton technique with multiple vector Jacobian (IQN-MVJ) and the multi-vector update quasi-Newton technique (MVQN). These coupling algorithms are differentiated based on whether the approximation of the Jacobian is performed for the entire black-box system (IQN-ILS and IQN-MVJ) or for both individual solvers (IBQN-LS and MVQN). Moreover, a distinction is made between methods which perform the approximation with either least-squares models (IQN-ILS and IBQN-LS) or multivector techniques (IQN-MVJ and MVQN). Their performance is compared by solving a 1D flexible tube case, using the in-house coupling software CoCoNuT. Both the memory usage and number of iterations between structure and flow solvers in each time step are examined. The techniques using a multi-vector approach require explicit matrix construction, so that memory requirements scale quadratically, whereas the least-squares techniques have a matrix-free implementation, resulting in linear scaling. In terms of convergence they are comparable.

A Robin‐Neumann scheme with quasi‐Newton acceleration for partitioned fluid‐structure interaction

“…Solving the same least squares problem as in Equation ( 13) for 𝜶 k , the implicit multi-vector least squares (IMVLS) variant allows to evaluate this expression without any explicit Jacobian representation. 17 Aside from the references already given, the works by Lindner et al 10 or Delaissé et al 45 are recommended for an overview of different IQN methods.…”

“…While this approach yields comparable convergence rates, the block quasi-Newton strategy requires the solution of two linear systems per coupling step and comes with a higher implementation effort, 9,49 in particular if the computational cost are supposed to grow linearly with the interface degrees of freedom to ensure the IQN method's efficiency for large-scale simulations. 17,45 Lastly, Bogaers et al did not study the quasi-Newton update's effect on the parameter-dependency of the artificial compressibility method.…”

A Robin-Neumann Scheme with Quasi-Newton Acceleration for Partitioned Fluid-Structure Interaction

“…Solving the same least squares problem as in Equation ( 13) for α k , the implicit multi-vector least squares (IMVLS) variant allows to evaluate this expression without any explicit Jacobian representation [16]. Aside from the references already given, the works by Lindner et al [9] or Delaissé et al [44] are recommended for an overview of different IQN methods.…”

On the number of subproblem iterations per coupling step in partitioned fluid-structure interaction simulations

“…They differ in when the interface data is updated, i.e., after only one of the solvers or after both, and in the technique employed to approximate the Jacobian, such as the least-squares or the multi-vector approach [14]. An extensive overview is given in [15].…”