In this work, the dual-weighted residual (DWR) method is applied to obtain a certified incremental proper orthogonal decomposition (POD) based reduced order model. A novel approach called MORe DWR (Model Order Rduction with Dual-Weighted Residual error estimates) is being introduced. It marries tensor-product space-time reduced-order modeling with time slabbing and an incremental POD basis generation with goal-oriented error control based on dual-weighted residual estimates. The error in the goal functional is being estimated during the simulation and the POD basis is being updated if the estimate exceeds a given threshold. This allows an adaptive enrichment of the POD basis in case of unforeseen changes in the solution behavior which is of high interest in many real-world applications. Consequently, the offline phase can be skipped, the reduced-order model is being solved directly with the POD basis extracted from the solution on the first time slab and -- if necessary -- the POD basis is being enriched on-the-fly during the simulation with high-fidelity finite element solutions. Therefore, the full-order model solves can be reduced to a minimum, which is demonstrated on numerical tests for the heat equation and elastodynamics.
翻译:MORe DWR:递增POD基础上面向目标的误差控制的空间时间约简模型。采用双重加权残差(DWR)方法获得认证的递增POD基础的基础上,引入了一种新的方法称为MORe DWR。它将张量积空间时间约简模型与时间片和基于双重加权残差估计的基于目标的误差控制相结合。在模拟期间估计目标函数的误差,如果估计值超过给定的阈值,则更新POD基础。这允许在解决方案行为出现未预料的变化的许多实际应用中对POD基础进行自适应强化。因此,可以跳过离线阶段,从第一个时间片的解中提取POD基础并直接求解约简模型,如果需要,在模拟期间通过高真实度的有限元解进行实时强化。因此,可以将全阶段模型的求解降至最小。这在热方程和弹性动力学的数值测试中得到了证明。