Numerical and physical modeling are the two main tools available for predicting the influence of water waves on coastal or offshore structures. Both models have their strengths and weaknesses. An integrated use of numerical and physical modeling which exploits their advantages can provide an optimal description of full-scale, realistic engineering problems. In this series of two papers, we report on a generalized three-dimensional (3D) deterministic coupling theory for multidirectional nonlinear wave propagation from a numerical model to a physical model up to second order. In this work, the second-order coupling theory developed by [Z. Yang, S. Liu, H.B. Bingham and J. Li, 2014a. Second-order coupling of numerical and physical wave tanks for 2D irregular waves. Part I Formulation, implementation and numerical properties. Coast. Eng. 92, 48–60] and the ad hoc unified wave generation theory developed by [H. Zhang, H.A., Schäffer, K.P., Jakobsen, 2007. Deterministic combination of numerical and physical coastal wave models. Coast. Eng. 54, 171–186.] have been extended to include the coupling between multidirectional nonlinear waves. In part I of this article series, the full second-order 3D coupling theory for multidirectional nonlinear waves is been described in detail. A novel generalized fully-nonlinear motion boundary equation has been derived, which allows the interface between the numerical and physical wave domains to be a 3D arbitrarily shaped wavemaker system. The corresponding 3D wave coupling equation is given for I-, L-, and O-shaped wavemaker layouts. The new formulation is presented in a unified form for an arbitrarily shaped layout of planar wavemakers, and covers both hinged and piston wavemaker types. For the second-order dispersive correction of the paddle stroke, the super-harmonic and sub-harmonic wave-wave interactions have been taken into account. For practical implementation, a typical discretization method for the 3D coupling equations is derived by considering the case of the I-shaped piston wavemaker. The new 3D coupling equations are solved by a combined five-point Lagrange interpolation and the fourth-order Runge–Kutta scheme. Numerical evaluations of the implementation have been conducted by considering a theoretical second-order unidirectional wave field over a range of spatial and time resolutions. The results thus obtained indicate that the proposed discretization scheme is accurate and effective. In addition, the precision of the discrete scheme is observed to be closely related to the spatial and temporal resolution. A separate experimental validation of the theory is presented in Part II.

数值和物理建模是可用于预测水浪对沿海或近海结构影响的两个主要工具。两种模式都有其优点和缺点。充分利用数值和物理模型的优势，可以全面，实际地解决工程问题。在这一系列的两篇论文中，我们报告了一种广义的三维（3D）确定性耦合理论，用于从数值模型到物理模型的多方向非线性波传播，直至二阶。在这项工作中，[Z。Yang，S.Liu，HB Bingham和J.Li，2014a。二维不规则波的数值和物理波箱的二阶耦合。第一部分配方，实施和数值特性。海岸。。92，48-60]和由[H。Zhang，HA，Schäffer，KP，Jakobsen，2007。数值和物理海浪模型的确定性组合。海岸。。54，171–186。]已扩展为包括多向非线性波之间的耦合。在本系列文章的第一部分中，详细描述了多向非线性波的完整二阶3D耦合理论。推导了一个新颖的广义的完全非线性运动边界方程，该方程使数值和物理波域之间的界面成为3D任意形状的波发生器系统。针对I形，L形和O形造波器布局，给出了相应的3D波耦合方程。新的公式以统一的形式呈现给平面波发生器的任意形状的布局，并涵盖了铰链和活塞式造波器两种类型。对于桨叶行程的二阶色散校正，已考虑了超谐波和次谐波波-波相互作用。对于实际的实现，通过考虑I形活塞波发生器的情况，得出了3D耦合方程式的典型离散化方法。新的3D耦合方程由五点Lagrange插值和四阶Runge-Kutta方案求解。通过在一定的空间和时间分辨率范围内考虑理论上的二阶单向波场，对实施方案进行了数值评估。由此获得的结果表明，所提出的离散化方案是准确且有效的。此外，观察到离散方案的精度与空间和时间分辨率密切相关。在第二部分中提供了对该理论的单独实验验证。