No abstract

No abstract

No abstract

No abstract

An approach combining proper orthogonal decomposition (POD) with linear regression, which is called gappy POD, is used to obtain complete flow solutions in steady aerodynamic applications from knowledge of a (suitable) POD basis and the solution at very few points. In aerodynamics the partial or gappy data can be gathered by wind tunnel experiments. The effectiveness of the Gappy POD will be demonstrated on an industrial testcase of the wing-body configuration MEGAFLUG. Proper Orthogonal DecompositionSuppose a set of m snapshots {u 1 , . . . , u m } is given. In our application, these snapshot vectors u i ∈ R d·n , i = 1, . . . , m contain d flow unknowns (e.g. pressure and density) at each of the n grid points of the computational mesh and are solutions to the Navier-Stokes equations. Each u i (α, M a) represents the steady flow for a different combination of the two parameters angle of attack α and Mach number M a. The goal of POD is to find a set of orthonormal basis vectorsis solved. In other words, a set of basis vectors {φ i } k i=1 for a k-dimensional subspace is sought such that the difference between the snapshots and their orthogonal projection onto the subspace is minimized. The solution to this problem is given by the left singular vectors of the snapshot matrix Gappy Proper Orthogonal DecompositionThe Gappy POD (GPOD) was first developed by Everson and Sirovich [2] in the context of reconstructing human face images. In [3] the gappy POD methodology was extended to fluid dynamic applications. The basic idea of GPOD is that the POD basis together with gappy data (which is data given at very few of the grid points) are used to reconstruct the flow vector for the entire grid. To understand the concept of gappyness, consider a flow vector u ∈ R n for one flow unknown, e.g. density or pressure. Each entry u j of this flow vector stands for the value of the flow unknown at the jth grid point. Suppose the gappy data is given at the grid points j 1 , . . . , jñ, whereñ is the number of points where data is given. By defining the selection or mask matrixwhere e j stands for the jth unit vector, only the known entries of a flow vector can be selected via P T ·u = u j1 · · · u jñ T .The goal of GPOD is to find a solution within the POD subspace, which can be expressed as Φ k · a with the POD basis Φ k and the coefficient vector a(α, M a) ∈ R k , such that it matches the data points as closely as possible. Hence, GPOD computes the vector a that satisfiesA solution to this so-called least squares or linear regression problem is given by a linear system of the form