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.

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