Numerical wave propagation and steady-state solutions - Soft wall and outer boundary conditions

For wave propagation in periodic media with strong nonlinearity, steady-state solutions can be obtained by solving a corresponding nonlinear delay differential equation (DDE). Based on the periodicity, the steady-state response of a repeated particle or segment in the media contains the complete information of solutions for the wave equation. Considering the motion of the selected particle or segment as a variable, motions of its adjacent particles or segments can be described by the same variable function with different phases, which are delayed variables. Thus, the governing equation for wave propagation can be converted to a nonlinear DDE with multiple delays. A modified incremental harmonic balance (IHB) method is presented here to solve the nonlinear DDE by introducing a delay matrix operator, where a direct approach is used to efficiently and automatically construct the Jacobian matrix for the nonlinear residual in the IHB method. This method is presented by solving an example of a one-dimensional monatomic chain under a nonlinear Hertzian contact law. Results are well matched with those in previous work, while calculation time and derivation effort are significantly reduced. Also there is no additional derivation required to solve new wave systems with different governing equations.

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REEF3D::FNPF: A Flexible Fully Nonlinear Potential Flow Solver

Volume 2: CFD and FSI ◽

10.1115/omae2019-96524 ◽

2019 ◽

Author(s):

Hans Bihs ◽

Weizhi Wang ◽

Tobias Martin ◽

Arun Kamath

Keyword(s):

Wave Propagation ◽

Free Surface ◽

Boundary Conditions ◽

Potential Flow ◽

Surface Boundary ◽

Flow Solver ◽

Surface Boundary Conditions ◽

Nonlinear Potential ◽

Numerical Wave ◽

Computational Resources

Abstract In situations where the calculation of ocean wave propagation and impact on offshore structures is required, fast numerical solvers are desired in order to find relevant wave events in a first step. After the identification of the relevant events, Computational Fluid Dynamics (CFD) based Numerical Wave Tanks (NWT) with an interface capturing two-phase flow approach can be used to resolve the complex wave structure interaction, including breaking wave kinematics. CFD models emphasize detail of the hydrodynamic physics, which makes them not the ideal candidate for the event identification due to the large computational resources involved. In the current paper a new numerical wave model is represented that solves the Laplace equation for the flow potential and the nonlinear kinematic and dynamics free surface boundary conditions. This approach requires reduced computational resources compared to CFD based NWTs. In contrast to existing approaches, the resulting fully nonlinear potential flow solver REEF3D::FNPF uses a σ-coordinate grid for the computations. Solid boundaries are incorporated through a ghost cell immersed boundary method. The free surface boundary conditions are discretized using fifth-order WENO finite difference methods and the third-order TVD Runge-Kutta scheme for time stepping. The Laplace equation for the potential is solved with Hypres stabilized bi-conjugated gradient solver preconditioned with geometric multi-grid. REEF3D::FNPF is fully parallelized following the domain decomposition strategy and the MPI communication protocol. The model is successfully tested for wave propagation benchmark cases for shallow water conditions with variable bottom as well as deep water.

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Modeling magnetized star-planet interactions: boundary conditions effects

Proceedings of the International Astronomical Union ◽

10.1017/s1743921313011162 ◽

2013 ◽

Vol 8 (S300) ◽

pp. 330-334 ◽

Cited By ~ 2

Author(s):

Antoine Strugarek ◽

Allan Sacha Brun ◽

Sean P. Matt ◽

Victor Reville

Keyword(s):

Steady State ◽

Boundary Conditions ◽

Numerical Simulations ◽

Stellar Wind ◽

Stellar Surface ◽

Numerical Studies ◽

Rotational Evolution ◽

Further Development ◽

Steady State Solutions

AbstractWe model the magnetized interaction between a star and a close-in planet (SPMIs), using global, magnetohydrodynamic numerical simulations. In this proceedings, we study the effects of the numerical boundary conditions at the stellar surface, where the stellar wind is driven, and in the planetary interior. We show that is it possible to design boundary conditions that are adequate to obtain physically realistic, steady-state solutions for cases with both magnetized and unmagnetized planets. This encourages further development of numerical studies, in order to better constrain and undersand SPMIs, as well as their effects on the star-planet rotational evolution.

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On a Continuum Theory for a Laminated Medium

Journal of Applied Mechanics ◽

10.1115/1.3423018 ◽

1973 ◽

Vol 40 (2) ◽

pp. 527-532 ◽

Cited By ~ 22

Author(s):

D. S. Drumheller ◽

A. Bedford

Keyword(s):

Wave Propagation ◽

Steady State ◽

Composite Materials ◽

Boundary Conditions ◽

Mode Shape ◽

Continuum Theory ◽

Higher Order ◽

Interface Boundary ◽

Interface Boundary Conditions

An extension of the effective stiffness theory developed for elastic laminates by Sun, Achenbach, and Herrmann [1] is presented in a form suitable for the solution of dynamical processes in composite materials including determination of stresses. A derivation of displacement and stress interface boundary conditions suitable for higher-order theories is presented. The theory is illustrated with dispersion and mode shape results for two examples of steady-state wave propagation.

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