Proper Orthogonal Decomposition (POD) plays an important role in the analysis of complex nonlinear systems governed by partial differential equations (PDEs), since it can describe the full-order system in a simplified but representative way using a handful of dominant dynamic modes. However, determining a POD from the results of complex unsteady simulations is often impractical using traditional approaches due to the need to store a large number of high-dimensional solutions. As an alternative, the incremental Singular Value Decomposition (SVD) has been developed, which can be used to avoid the storage problem by performing the POD analysis on the fly using a single-pass updating algorithm. Nevertheless, the total computing cost of incremental SVD is more than traditional approaches. In order to reduce this total cost, we incorporate POD mode truncation into the incremental procedure, leading to an enhanced algorithm for incremental SVD. Two error estimators are formulated for this enhanced incremental SVD based on an aggregated expression of the snapshot solutions, equipping the proposed algorithm with criteria for choosing the truncation number. The effectiveness of these estimators and the parallel efficiency of the enhanced algorithm are demonstrated using transient solutions from representative model problems. Numerical results show that the enhanced algorithm can significantly improve the computing efficiency for different kinds of datasets, and that the proposed algorithm is scalable in both the strong and weak sense.

Adaptive mesh refinement (AMR) is potentially an effective way to automatically generate computational meshes for high-fidelity simulations such as Large Eddy Simulation (LES). Adjoint methods, which are able to localize error contributions, can be used to optimize the mesh for computing a physical quantity of interest (e.g. lift, drag) during AMR. When adjoint-based AMR techniques are applied to LES, primal flow solutions are needed to solve the adjoint problem backward in time due to the nonlinearity of Navier-Stokes equations. However, the resources required to store primal flow solutions can be huge, even prohibitive, in practical problems because of the long averaging time for computing statistical quantities. In this paper, a Reduced-Order Model (ROM) based upon Proper Orthogonal Decomposition (POD) is introduced to circumvent this issue. First, an adjoint-based error estimation procedure is verified using a manufactured solution. Then a ROM-driven AMR strategy is studied using a LES model problem based on the 1D unsteady Burgers equation. Numerical results demonstrate that using ROMs not only lowers storage requirements, but also has no impact on the effectiveness of adjoint-based AMR.

Adaptive Mesh Refinement (AMR) is potentially an effective way to automatically generate computational meshes for high-fidelity simulations such as Large Eddy Simulation (LES). When combined with adjoint methods, which are able to localize error contributions, AMR can generate meshes that are optimal for computing a physical quantity of interest (e.g. lift or drag). In order to apply adjoint-based AMR techniques to LES, primal flow solutions are needed to solve the adjoint problem backward in time. However, the resources required to store primal flow solutions can be huge, even prohibitive, in practical problems because of the typically very fine meshes and long averaging times for computing the statistical quantities of interest. Here, a Reduced-Order Representation (ROR) based upon proper orthogonal decomposition is introduced to address this issue. We develop an Enhanced Online Algorithm (EOA) based on incremental singular value decomposition to build this ROR online, which makes adjoint-based AMR feasible for practical applications. An adjoint-based error estimation procedure is first introduced, and verified using a manufactured solution. Then a ROR-driven AMR strategy is studied using a 1D unsteady Burgers problem with a multi-frequency forcing term. This is also used to evaluate the EOA for ROR-driven AMR. Numerical results demonstrate that the enhanced online algorithm generates RORs that are sufficiently accurate for AMR, avoiding the storage of almost all of the primal solution data.