A continuous Galerkin finite element method that allows mixed boundary conditions without the need for Lagrange multipliers or user-defined parameters is developed. A mixed coupling of Lagrange and Raviart-Thomas basis functions are used. The method is proven to have a Hamiltonian-conserving spatial discretisation and a symplectic time discretisation. The energy residual is therefore guaranteed to be bounded for general problems and exactly conserved for linear problems. The linear 2D wave equation is discretised and modelled by making use of a port-Hamiltonian framework. This model is verified against an analytic solution and shown to have standard order of convergence for the temporal and spatial discretisation. The error growth over time is shown to grow linearly for this symplectic method, which agrees with theoretical results. A modal analysis is performed which verifies that the eigenvalues of the model accurately converge to the exact eigenvalues, as the mesh is refined. The port-Hamiltonian framework allows boundary coupling with bond-graph or, more generally, lumped parameter models, therefore unifying the two fields of lumped parameter modelling and continuum modelling of Hamiltonian systems. The wave domain discretisation is shown to be equivalent to a coupling of canonical port-Hamiltonian forms. This feature allows the model to have mixed boundary conditions as well as to have mixed causality interconnections with other port-Hamiltonian models. A model of the 2D wave equation is coupled, in a monolithic manner, with a lumped parameter model of an electromechanical linear actuator. The combined model is also verified to conserve energy exactly.

开发了一种允许混合边界条件而无需拉格朗日乘子或用户定义参数的连续 Galerkin 有限元方法。使用了拉格朗日基函数和 Raviart-Thomas 基函数的混合耦合。该方法被证明具有哈密顿量守恒空间离散化和辛时间离散化。因此，保证能量残差对于一般问题是有界的，而对于线性问题是完全守恒的。线性二维波动方程通过使用 port-Hamiltonian 框架进行离散化和建模。该模型针对解析解进行了验证，并显示出具有时间和空间离散化的标准收敛顺序。对于这种辛方法，误差随时间的增长呈线性增长，这与理论结果一致。执行模态分析以验证模型的特征值是否在网格细化时准确收敛到准确的特征值。port-Hamiltonian 框架允许与键图或更一般的集总参数模型进行边界耦合，因此统一了哈密顿系统的集总参数建模和连续介质建模这两个领域。波域离散化被证明等效于典型端口-汉密尔顿形式的耦合。此功能允许模型具有混合边界条件以及与其他端口-汉密尔顿模型具有混合因果关系。二维波动方程模型以整体方式与机电线性致动器的集总参数模型耦合。还验证了组合模型可以精确地节约能量。