Progressive construction of a parametric reduced-order model for PDE-constrained optimization

An adaptive approach to using reduced-order models as surrogates in PDE-constrained optimization is introduced that breaks the traditional offline-online framework of model order reduction. A sequence of optimization problems constrained by a given Reduced-Order Model (ROM) is defined with the goal of converging to the solution of a given PDE-constrained optimization problem. For each reduced optimization problem, the constraining ROM is trained from sampling the High-Dimensional Model (HDM) at the solution of some of the previous problems in the sequence. The reduced optimization problems are equipped with a nonlinear trust-region based on a residual error indicator to keep the optimization trajectory in a region of the parameter space where the ROM is accurate. A technique for incorporating sensitivities into a Reduced-Order Basis (ROB) is also presented, along with a methodology for computing sensitivities of the reduced-order model that minimizes the distance to the corresponding HDM sensitivity, in a suitable norm. The proposed reduced optimization framework is applied to subsonic aerodynamic shape optimization and shown to reduce the number of queries to the HDM by a factor of 4-5, compared to the optimization problem solved using only the HDM, with errors in the optimal solution far less than 0.1%

The GNAT method for nonlinear model reduction: effective implementation and application to computational fluid dynamics and turbulent flows

The Gauss--Newton with approximated tensors (GNAT) method is a nonlinear model reduction method that operates on fully discretized computational models. It achieves dimension reduction by a Petrov--Galerkin projection associated with residual minimization; it delivers computational efficency by a hyper-reduction procedure based on the `gappy POD' technique. Originally presented in Ref. [1], where it was applied to implicit nonlinear structural-dynamics models, this method is further developed here and applied to the solution of a benchmark turbulent viscous flow problem. To begin, this paper develops global state-space error bounds that justify the method's design and highlight its advantages in terms of minimizing components of these error bounds. Next, the paper introduces a `sample mesh' concept that enables a distributed, computationally efficient implementation of the GNAT method in finite-volume-based computational-fluid-dynamics (CFD) codes. The suitability of GNAT for parameterized problems is highlighted with the solution of an academic problem featuring moving discontinuities. Finally, the capability of this method to reduce by orders of magnitude the core-hours required for large-scale CFD computations, while preserving accuracy, is demonstrated with the simulation of turbulent flow over the Ahmed body. For an instance of this benchmark problem with over 17 million degrees of freedom, GNAT outperforms several other nonlinear model-reduction methods, reduces the required computational resources by more than two orders of magnitude, and delivers a solution that differs by less than 1% from its high-dimensional counterpart

Solution of Heat Transfer and Fluid Flow problems using meshless Radial Basis Function method

In the past, the world of numerical solutions for Heat Transfer and Fluid Flow problems has been dominated by Finite Element Method, Finite Difference Method, Finite Volume Method, and more recently the Boundary Element Method. These methods revolve around using a mesh or grid to solve problems. However, problems with irregular boundaries and domains can be difficult to properly discretize; In this thesis, heat transfer and fluid flow problems are solved using Radial Basis Functions. This method is meshless, easy to understand, and even easier to implement. Radial Basis Functions are used to solve lid-driven cavity flow, natural convection in a square enclosure, flow with forced convection over backward facing step and flow over an airfoil. Codes are developed using MATLAB. The results are compared with COMSOL and FLUENT, two popular commercial codes widely used. COMSOL is a finite element model while FLUENT is a finite volume-based code

Data-driven Balanced Truncation for Predictive Model Order Reduction of Aeroacoustic Response

Rapid prediction of the aeroacoustic response is a key component in the design of aircraft and turbomachinery. While it is possible to achieve accurate predictions using direct solution of the compressible Navier-Stokes equations, applications of such solvers is not feasible in design optimization due to the high cost of resolving wave phenomena in an Eulerian setting. In this work, we propose a technique for highly accelerated predictions of aeroacoustic response using a data-driven model reduction approach based on the eigensystem realization algorithm (ERA), as a non-intrusive balanced truncation method. Specifically, we create and compare ERA ROMs based on the training data generated by solving the linearized and nonlinear Euler equations with Gaussian pulse inputs, and use them for prediction of the aeroacoustic response of an airfoil subject to different types of gust loading. The results show that both ROMs are in good agreement with the full-order model (FOM) solution in a purely predictive setting, while achieving orders of magnitude reduction in the online computation time. Using ERA for prediction of the acoustic response requires activating each input channel separately in the FOM for training ROMs, and operating on a large Hankel matrix, that can become computationally infeasible. We address this bottleneck in two steps: first, we propose a multi-fidelity gappy POD method to identify the most impactful input channels based on a coarser grid. Therefore, we reduce the computation cost on the FOM and ROM levels as we build the Markov sequence by querying the high-resolution FOM only for the input channels identified by gappy POD. Second, we use tangential interpolation at the ROM level to reduce the size of the Hankel matrix. The proposed methods enable application of ERA for highly accurate online acoustic response prediction and reduce the offline computation cost of ROMs

Development of CAE tools for fluid-structure interaction and erosion in turbomachinery virtual prototyping

The work presented in this thesis is based on the development of advanced computer aided engineering tools dedicated to multi-physics coupled problems. Starting from the state of the art of numerical tools used in virtual prototyping and testing of turbomachinery systems, we found two interesting and actual possible developments focused on the improved implementation of fluid-structure interaction and material wearing solvers. For both the topics we will present a brief overview with the contextualization on the industrial and research state of the art, the detailed description of mathematical models (Chapter 2), discretized (FEM) stabilized formulations, time integration schemes and coupling algorithms used in the implementation (Chapter 3). The second part of the thesis (Chapter 4-7) will report some application of the developed tools on some latest challenges in turbomachinery field as rain erosion and load control in wind turbines and non-linear aeroelasticity in large axial fans

SOLID-SHELL FINITE ELEMENT MODELS FOR EXPLICIT SIMULATIONS OF CRACK PROPAGATION IN THIN STRUCTURES

Crack propagation in thin shell structures due to cutting is conveniently simulated using explicit finite element approaches, in view of the high nonlinearity of the problem. Solidshell elements are usually preferred for the discretization in the presence of complex material behavior and degradation phenomena such as delamination, since they allow for a correct representation of the thickness geometry. However, in solid-shell elements the small thickness leads to a very high maximum eigenfrequency, which imply very small stable time-steps. A new selective mass scaling technique is proposed to increase the time-step size without affecting accuracy. New ”directional” cohesive interface elements are used in conjunction with selective mass scaling to account for the interaction with a sharp blade in cutting processes of thin ductile shells

Adaptive computational methods for aerothermal heating analysis

The development of adaptive gridding techniques for finite-element analysis of fluid dynamics equations is described. The developmental work was done with the Euler equations with concentration on shock and inviscid flow field capturing. Ultimately this methodology is to be applied to a viscous analysis for the purpose of predicting accurate aerothermal loads on complex shapes subjected to high speed flow environments. The development of local error estimate strategies as a basis for refinement strategies is discussed, as well as the refinement strategies themselves. The application of the strategies to triangular elements and a finite-element flux-corrected-transport numerical scheme are presented. The implementation of these strategies in the GIM/PAGE code for 2-D and 3-D applications is documented and demonstrated