Gianluca Geraci

TSI metamodels-based multi-objective robust optimization

Purpose – This paper aims to deal with an efficient strategy for robust optimization when a large number of uncertainties are taken into account. Design/methodology/approach – ANOVA analysis is used in order to perform a variance-based decomposition and to reduce stochastic dimension based on an appropriate criterion. A massive use of metamodels allows reconstructing response surfaces for sensitivity indexes in the design variables plan. To validate the proposed approach, a simplified configuration, an inverse problem on a 1D nozzle flow, is solved and the performances compared to an exact Monte Carlo reference solution. Then, the same approach is applied to the robust optimization of a turbine cascade for thermodynamically complex flows. Findings – First, when the stochastic dimension is reduced, the error on the variance between the reduced and the complete problem was found to be roughly estimated by the quantity (1−T¯ TSI )×100, where T¯ TSI is the summation of TSI concerning the variables respecting ...

A Novel Weakly-Intrusive Non-linear Multiresolution Framework for Uncertainty Quantification in Hyperbolic Partial Differential Equations

In this paper, a novel multiresolution framework, namely the Truncate and Encode (TE) approach, previously proposed by some of the authors (Abgrall et al. in J Comput Phys 257:19–56, 2014. doi:10.1016/j.jcp.2013.08.006), is generalized and extended for taking into account uncertainty in partial differential equations (PDEs). Innovative ingredients are given by an algorithm permitting to recover the multiresolution representation without requiring the fully resolved solution, the possibility to treat a whatever form of pdf and the use of high-order (even non-linear, i.e. data-dependent) reconstruction in the stochastic space. Moreover, the spatial-TE method is introduced, which is a weakly intrusive scheme for uncertainty quantification (UQ), that couples the physical and stochastic spaces by minimizing the computational cost for PDEs. The proposed scheme is particularly attractive when treating moving discontinuities (such as shock waves in compressible flows), even if they appear during the simulations as it is common in unsteady aerodynamics applications. The proposed method is very flexible since it can easily coupled with different deterministic schemes, even with high-resolution features. Flexibility and performances of the present method are demonstrated on various numerical test cases (algebraic functions and ordinary differential equations), including partial differential equations, both linear and non-linear, in presence of randomness. The efficiency of the proposed strategy for solving stochastic linear advection and Burgers equation is shown by comparison with some classical techniques for UQ, namely Monte Carlo or the non-intrusive polynomial chaos methods.

Stochastic Discrete Equation Method (sDEM) for two-phase flows

A new scheme for the numerical approximation of a five-equation model taking into account Uncertainty Quantification (UQ) is presented. In particular, the Discrete Equation Method (DEM) for the discretization of the five-equation model is modified for including a formulation based on the adaptive Semi-Intrusive (aSI) scheme, thus yielding a new intrusive scheme (sDEM) for simulating stochastic two-phase flows. Some reference test-cases are performed in order to demonstrate the convergence properties and the efficiency of the overall scheme. The propagation of initial conditions uncertainties is evaluated in terms of mean and variance of several thermodynamic properties of the two phases.

Multi-objective Design Optimization Using High-Order Statistics for CFD Applications

This work illustrates a practical and efficient method for performing multi-objective optimization using high-order statistics. It is based on a Polynomial Chaos framework, and evolutionary algorithms. In particular, the interest of considering high-order statistics for reducing the number of uncertainties is studied. The feasibility of the proposed method is proved on a Computational Fluid-Dynamics (CFD) real-case application.

Node-pair finite volume/finite element schemes for the Euler equation in cylindrical and spherical coordinates

A numerical scheme is presented for the solution of the compressible Euler equations in both cylindrical and spherical coordinates. The unstructured grid solver is based on a mixed finite volume/finite element approach. Equivalence conditions linking the node-centered finite volume and the linear Lagrangian finite element scheme over unstructured grids are reported and used to devise a common framework for solving the discrete Euler equations in both the cylindrical and the spherical reference systems. Numerical simulations are presented for the explosion and implosion problems with spherical symmetry, which are solved in both the axial-radial cylindrical coordinates and the radial-azimuthal spherical coordinates. Numerical results are found to be in good agreement with one-dimensional simulations over a fine mesh.

An Alternative Formulation for Design Under Uncertainty

A novel formulation for design under uncertainty is presented, which is based on the computation of the mean value and the minimum of the function. The aim of the method is to exert a stronger control on the system output variability in the optimization loop at a moderate cost. This would reduce post-processing analysis of the PDF of the resulting optimal designs, by converging rapidly to the interesting individuals. In other words, in the set of designs resulting from the optimization, the new approach should be capable of discarding poor-performance design. Also, no a priori assumption of optimal PDF is made. The preliminary results presented in the paper proves the benefit of the new formulation.

Towards a unified multiresolution scheme for treating discontinuities in differential equations with uncertainties

In the present work, a method for solving partial differential equations with uncertainties is presented. A multiresolution method, permitting to compute statistics for the entire solution and in presence of a whatever form of the probability density function, is extended to perform an adaptation in both physical and stochastic spaces. The efficiency of this strategy, in terms of refinement/coarsening capabilities, is demonstrated on several test-cases by comparing with respect to other more classical techniques, namely Monte Carlo (MC) and Polynomial Chaos (PC). Finally, the proposed strategy is applied to the heat equation showing very promising results in terms of accuracy, convergence and regularity.

Polynomial chaos assessment of design tolerances for vortex-induced vibrations of two cylinders in tandem

Abstract The presence of aerodynamics loadings makes the design of some classes of elastic structures, as, for instance, marine structures and risers, very challenging. Moreover, capturing the complex physical interaction between the structure and the fluid is challenging for both theoretical and numerical models. One of the most important phenomena that appear in these situations is vortex-induced vibrations. The picture is even more complicated when multiple elastic elements are close enough to interact by modifying the fluid flow pattern. In the present work, we show how the common design practice for these structures, which is entirely based on deterministic simulations, needs to be complemented by the uncertainty quantification analysis. The model problem is a structure constituted by two elastically mounted cylinders exposed to a two-dimensional uniform flow at Reynolds number 200. The presence of a manufacturing tolerance in the relative position of the two cylinders, which we consider to be a source of uncertainty, is addressed. The overall numerical procedure is based on a Navier–Stokes immersed boundary solver that uses a flexible moving least squares approach to compute the aerodynamics loadings on the structure, whereas the uncertainty quantification propagation is obtained by means of a nonintrusive polynomial chaos technique. A range of reduced velocities is considered, and the quantification, in a probabilistic sense, of the difference in the performances of this structure with respect to the case of an isolated cylinder is provided. The numerical investigation is also complemented by a global sensitivity analysis based on the analysis of variance.

Equivalence conditions for the finite volume and finite element methods in spherical coordinates

A numerical technique for the solution of the compressible flow equations over unstructured grids in a spherical reference is presented. The proposed approach is based on a mixed finite volume/finite element discretization in space. Equivalence conditions relating the finite volume and the finite element metrics in spherical coordinates are derived. Numerical simulations of the explosion and implosion problems for inviscid compressible flows are carried out to evaluate the correctness of the numerical scheme and compare fairly well to one-dimensional simulations over very fine grids.