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6.3.2 Handling Boundary Conditions One of the key issues in reduced-order modeling is the incorporation of boundary conditions. If the boundary conditions are homogeneous, no changes are required in the aforementioned methodologies of obtaining POD basis functions and final reduced-order model. However, incorporating non-homogeneous Dirichlet and Neumann boundary conditions is non-trivial in general. Gunzburger et al. [89] have developed a generic framework to incorporate boundary conditions in the POD-based ROMs. For non-homogeneous Dirichlet boundary conditions, we utilize the idea of computing POD basis elements for fluctuations around the mean value of the snapshots. Given Nt snapshots, first we compute the mean value of the snapshots y = (1/Nt)Nt yj. Next, the snapshot j=1 matrix is modified as Y = {y1 − y, . . . , yNt − y}, and POD basis functions are computed using this modified input ensemble. This helps in projecting out the boundary condition to the mean value of the snapshot and allows POD basis functions to follow a homogeneous boundary condition. For ROM purposes, a state variable is now expressed as M i=1 Let the boundary conditions be y(0,t) = A, y(L,t) = B. Since all the snapshots satisfy this, y(x) will also satisfy this, i.e., y(0) = A and y(L) = B. Hence, φi(0) = φi(L) = 0, i = 1, . . . , M , which helps in ensuring that the boundary conditions are always satisfied by the solution obtained after integrating ROM. In some cases, especially when a boundary condition appears as a decision variable in an optimization problem, we desire explicit occurence of the boundary condition in the expansion (6.17). Let such a boundary condition be yb. In this case, first we subtract yb from all the snapshots at all spatial points {xi}Nx . To ensure consistency i=1 y(x, t) = y(x) + ai(t)φi(x) (6.17) for the other boundary condition, mean value of the snapshots is then computed for these modified snapshots, i.e., y = (1/N ) Nt (yj − yb). Consequently, the snapshot matrix used b tj=1 to compute POD basis set becomes Y = {y1 −yb −yb, . . . , yNt −yb −yb}. As a result, expansion 6.3 Reduced-order Modeling Chapter 6. Reduced-order Modeling for Optimization 107PDF Image | Design and Operation of Pressure Swing Adsorption Processes
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