In the present work, a new methodology is proposed for building surrogate parametric models of engineering systems based on modular assembly of pre-solved modules. Each module is a generic parametric solution considering parametric geometry, material and boundary conditions. By assembling these modules and satisfying continuity constraints at the interfaces, a parametric surrogate model of the full problem can be obtained. In the present paper, the PGD technique in connection with NURBS geometry representation is used to create a parametric model for each module. In this technique, the NURBS objects allow to map the governing boundary value problem from a parametric non-regular domain into a regular reference domain and the PGD is used to create a reduced model in the reference domain. In the assembly stage, an optimization problem is solved to satisfy the continuity constraints at the interfaces. The proposed procedure is based on the offline–online paradigm: the offline stage consists of creating multiple pre-solved modules which can be afterwards assembled in almost real-time during the online stage, enabling quick evaluations of the full system response. To show the potential of the proposed approach some numerical examples in heat conduction and structural plates under bending are presented.

中文翻译：

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模块化参数 PGD 支持在线求解偏微分方程


在这项工作中，提出了一种基于预求解模块的模块化组装构建工程系统代理参数模型的新方法。每个模块都是考虑参数化几何、材料和边界条件的通用参数化解。通过组装这些模块并满足界面处的连续性约束，可以获得完整问题的参数化代理模型。在本文中，PGD 技术与 NURBS 几何表示相结合，用于为每个模块创建参数化模型。在这种技术中，NURBS 对象允许将控制边界值问题从参数化非正则域映射到常规参考域，并且 PGD 用于在参考域中创建简化模型。在装配阶段，求解优化问题以满足界面处的连续性约束。所提出的程序基于离线-在线范式：离线阶段包括创建多个预求解模块，这些模块随后可以在在线阶段几乎实时地组装，从而能够快速评估整个系统的响应。为了展示所提出的方法的潜力，提出了一些热传导和弯曲结构板的数值示例。