Efthymios N. Karatzas

Efthymios N. Karatzas, Giovanni Stabile, Leo Nouveau et al.

We propose a model order reduction technique integrating the Shifted Boundary Method (SBM) with a POD-Galerkin strategy. This approach allows to treat more complex parametrized domains in an efficient and straightforward way. The impact of the proposed approach is threefold. First, problems involving parametrizations of complex geometrical shapes and/or large domain deformations can be efficiently solved at full-order by means of the SBM, an unfitted boundary method that avoids remeshing and the tedious handling of cut cells by introducing an approximate surrogate boundary. Second, the computational effort is further reduced by the development of a reduced order model (ROM) technique based on a POD-Galerkin approach. Third, the SBM provides a smooth mapping from the true to the surrogate domain, and for this reason, the stability and performance of the reduced order basis are enhanced. This feature is the net result of the combination of the proposed ROM approach and the SBM. Similarly, the combination of the SBM with a projection-based ROM gives the great advantage of an easy and fast to implement algorithm considering geometrical parametrization with large deformations. The transformation of each geometry to a reference geometry (morphing) is in fact not required. These combined advantages will allow the solution of PDE problems more efficiently. We illustrate the performance of this approach on a number of two-dimensional Stokes flow problems.

Efthymios N. Karatzas

We develop and analyze a posteriori error estimators for a proper orthogonal decomposition-discrete empirical interpolation method (Pod-Deim) reduced order model applied to a parametric Poisson equation posed on a parameter-dependent domain defined by a level-set function. The full-order discretisations employ a cut finite element method (Cutfem) with Nitsche boundary conditions and ghost-penalty stabilization. Three complementary estimators are proposed: (i) Deim approximation quality indicators for the stiffness matrix and force vector, which are constant in the number of Pod modes, (ii) dual-norm residual estimators in both plain and Jacobi-preconditioned form, and (iii) a Pod tail-energy indicator. A rigorous theoretical framework is established, comprising a uniform coercivity result for the Cutfem bilinear form, an active-dof residual bound that accounts for ghost-penalty degrees of freedom, a combined a posteriori bound, and sharp effectivity analysis for the residual estimators. The key theoretical finding is that the large observed effectivity indices are explained by ghost-penalty degree-of-freedom inflation, and that restricting the residual to active degrees of freedom is predicted to reduce effectivity. Numerical experiments on a parametric ellipse domain with semi-axes confirm the theoretical predictions, achieve significant online speedup, and demonstrate algebraic convergence of the true error alongside exponential decay of the residual estimators.