Fast and multiscale formation of isogeometric matrices of microstructured geometric models

The matrix formation associated to high-order discretizations is known to be numerically demanding. Based on the existing procedure of interpolation and lookup, we design a multiscale assembly procedure to reduce the exorbitant assembly time in the context of isogeometric linear elasticity of complex microstructured geometries modeled via spline compositions. The developed isogeometric approach involves a polynomial approximation occurring at the macro-scale and the use of lookup tables with pre-computed integrals incorporating the micro-scale information. We provide theoretical insights and numerical examples to investigate the performance of the procedure. The strategy turns out to be of great interest not only to form finite element operators but also to compute other quantities in a fast manner as for instance sensitivity analyses commonly used in design optimization.

Model Order Reduction by geometrical parametrization for shape optimization in computational fluid dynamics

Andrea Manzoni, Gianluigi Rozza

Shape Optimization problems governed by partial differential equations result from many applications in computational fluid dynamics; they involve the repetitive evaluation of outputs expressed as functionals of the field variables and usually imply big computational efforts. For this reason looking for computational efficiency in numerical methods and algorithms is mandatory. The interplay between scientific computing and new reduction strategies is crucial in applications of great complexity. In order to achieve an efficient model order reduction, reduced basis methods built upon a high-fidelity ``truth'' finite element approximation -- and combined with suitable geometrical parametrization techniques for efficient shape description -- can be introduced, thus decreasing both the computational effort and the geometrical complexity. Starting from an excursus on classical approaches -- such as local boundary variation and shape boundary parametrization -- we focus on more efficient parametrization techniques which are well suited for a combination with a reduced basis approach, such as the one based on affine mapping (even automatic), nonaffine mapping (coupled with a suitable empirical interpolation technique for better numerical performances) and free-form deformations. We thus describe (and compare) the principal features of these parametrization techniques by showing some applications dealing with shape optimization of parametrized configurations in viscous flows,and discussing computational advantages and efficiency obtained by geometrical and computational model order reduction.

2010

Chebyshev interpolation for nonlinear eigenvalue problems

Peter Cedric Effenberger, Daniel Kressner

This work is concerned with numerical methods for matrix eigenvalue problems that are nonlinear in the eigenvalue parameter. In particular, we focus on eigenvalue problems for which the evaluation of the matrix-valued function is computationally expensive. Such problems arise, e.g., from boundary integral formulations of elliptic PDE eigenvalue problems and typically exclude the use of established nonlinear eigenvalue solvers. Instead, we propose the use of polynomial approximation combined with non-monomial linearizations. Our approach is intended for situations where the eigenvalues of interest are located on the real line or, more generally, on a pre-specified curve in the complex plane. A first-order perturbation analysis for nonlinear eigenvalue problems is performed. Combined with an approximation result for Chebyshev interpolation, this shows exponential convergence of the obtained eigenvalue approximations with respect to the degree of the approximating polynomial. Preliminary numerical experiments demonstrate the viability of the approach in the context of boundary element methods.

Springer Netherlands2012

Fluid-structure interaction for vascular flows: From supercomputers to laptops

Claudia Maria Colciago, Simone Deparis, Davide Forti

Several models exist for the simulation of vascular flows; they span from simple circuit models to full three-dimensional ones that take into account detailed features of the blood and of the arterialwall. Eachmodel comeswith both benefits and drawbacks, the main denominator being a compromise between detailed resolution requirements versus computational time. We first present a fluid-structure interaction computationalmodelwhere both the fluid and the structure are three dimensional. In particular, the fluid includes modeling of large eddies by the variationalmultiscalemethod. After time and space discretizations carried out by finite differences and finite elements, respectively, we set up a parallel solver based ondomain decomposition and a FaCSI preconditioner. These simulations allow one to capture details of the flow dynamics and of the structure deformation even in the transitional regime characterizing hemodynamics in the aorta. It takes roughly 10 hours to complete a simulation of one heartbeat with 35 million degrees of freedom on 2048 cores. We then reduce both the model and its numerical complexity. The structural model is simplified to a two-dimensional membrane located at the fluid-structure interface and the fluid computational domain is fixed. For a fixed geometry andmesh, these assumptions allow one to apply proper orthogonal decomposition and generate a space discretization which has only a few dozen degrees of freedom. It is then possible to perform the simulation of one heartbeat on a laptop in less than one second. Themodeling and numerical reduction therefore allows a dramatic reduction of computational time. However, the price to pay comes, on the one hand, in terms of the preparation of a reduced basis specific to the patient and the geometry of the vessel and, on the other hand, with a detriment of certain quantities of interest. For example, when using a finite element discretization with 9 million degrees of freedom, the offline part takes about 12 hours on 720 cores for the example provided in this work; in this case, the flow profiles in the aorta are pretty close to the full three-dimensional

De Gruyter2017

Fluid-structure interaction for vascular flows: From supercomputers to laptops

Claudia Maria Colciago, Simone Deparis, Davide Forti

De Gruyter2017