Model order reduction for large-scale structures with local nonlinearities

In solid mechanics, linear structures often exhibit (local) nonlinear behavior when close to failure. For instance, the elastic deformation of a structure becomes plastic after being deformed beyond recovery. To properly assess such problems in a real-life application, we need fast and multi-query evaluations of coupled linear and nonlinear structural systems, whose approximations are not straight forward and often computationally expensive. In this work, we propose a linear-nonlinear domain decomposition, where the two systems are coupled through the solutions on the linear-nonlinear interface. After necessary sensitivity analysis, e.g. for structures with a high dimensional parameter space, we adopt a non-intrusive method, e.g. Gaussian processes regression (GPR), to solve for the solution on the interface. We then utilize different model order reduction techniques to address the linear and nonlinear problems individually. To accelerate the approximation, we employ again the non-intrusive GPR for the nonlinearity, while intrusive model order reduction methods, e.g. the conventional reduced basis (RB) method or the static-condensation reduced- basis-element (SCRBE) method, are employed for the solution in the linear subdomain. We provide several numerical examples to demonstrate the effectiveness of our method.

Reduced order modeling techniques for numerical homogenization methods applied to linear and nonlinear multiscale problems

Yun Bai

The characteristic of effective properties of physical processes in heterogeneous media is a basic modeling and computational problem for many applications. As standard numerical discretization of such multiscale problems (e.g. with classical finite element method (FEM)) is often computationally prohibitive, there is a need for a novel computational algorithm able to capture the effective behavior of the physical system without resolving the finest scale in the system on the whole computational domain. In this thesis we propose and analyze a new class of numerical methods that combine the so-called finite element heterogeneous multiscale method (FE-HMM) with reduced order modeling techniques for linear and nonlinear multiscale problems. In the first part of the thesis we generalize the FE-HMM to elliptic problems with an arbitrary number of well-separated scales. We provide a rigorous a priori error analysis of this method that generalizes previous work restricted to two-scale problems. In the second part of the thesis, we develop our new reduced order multiscale method that combines the FE-HMM with reduced basis (RB) method. This method, the reduced basis finite element heterogeneous multiscale method (RB-FE-HMM) provides a significant improvement in computational efficiency compared to the FE-HMM, specially for high dimensional problems or high order methods. A priori and a posteriori error analyses are derived for linear elliptic problems, as well as goal oriented adaptivity techniques. The RB-FE-HMM is then generalized to a class of nonlinear elliptic and parabolic problems. A priori error analysis and extensive computational results for nonlinear problems are also provided.

EPFL2013

Numerical methods for structural anomaly detection using model order reduction and data-driven techniques

Caterina Bigoni

The objective of this thesis is to provide a mathematical and computational framework for the proactive maintenance of complex systems with a particular application to structural health monitoring (SHM). SHM techniques rely primarily on sensor responses to assess the risk associated with a structure of interest and seek to provide a support for automated decision-making strategy. An efficient integration of experimental measurements and numerical models is needed to exhaustively describe environmental and operational scenarios that a structure undergoes during its life time. We propose a simulation-based approach that combines the solution of a parametric time-dependent partial differential equation (PDE) for multiple input parameters with data-driven techniques to discriminate between healthy and damaged configurations. This process exploits an offline-online decomposition of tasks. The dataset of synthetic sensor measurements is generated offline by repeatedly solving a parametric PDE for a predefined configuration under several healthy variations. A reduced order model is employed to overcome the computational bottleneck associated with the many-query context by building a projection-based reduced basis method in combination with a Laplace-domain solver. Weeks method is adopted to numerically invert the Laplace transform and transform the signals to the time domain. The datasets are used to train various one-class machine learning algorithms in a semi-supervised setting, sensor by sensor. Finally, the outputs of the classifiers are used to assess the state of damage of the structure online. Using a decision-level fusion strategy, we provide insight on the existence, location, and severity of possible damages. The performance of SHM depends critically on the quality of the sensor measurements although their availability is often limited due to budget constraints and installation difficulties. We therefore propose a strategy to systematically place a fixed number of sensors on a structure of interest to minimize uncertainty at unsensed locations. The so-called inducing points, an outcome of sparse Gaussian processes, originally introduced to overcome the computational burden associated with performing a regression task with standard Gaussian processes, are here used to guide the sensor placement. A clustering approach is employed to select the sensor locations among a set of inducing points, computed for different input parameters. We apply this methodology to 2D and 3D problems to mimic the vibrational behavior of complex structures under the effect of an active source and show the effectiveness of the approach for (i) detecting damaged geometries and (ii) identifying the locations for a network of sensors. This framework considers the realistic case where damage types and locations are a priori unknown, thus overcoming the main limitation of existing simulation-based damage detection and sensor placement strategies for SHM.

EPFL2020

Structure-preserving approaches and data-driven closure modeling for model order reduction

Nicolò Ripamonti

In this thesis, we propose model order reduction techniques for high-dimensional PDEs that preserve structures of the original problems and develop a closure modeling framework leveraging the Mori-Zwanzig formalism and recurrent neural networks. Since high-fidelity approximations of PDEs often result in a large number of degrees of freedom, the need for iterative evaluations for numerical optimizations and rapid feedback is computationally challenging.The first part of this thesis is devoted to conserving the high-dimensional equation's invariants, symmetries, and structures during the reduction process. Traditional reduction techniques are not guaranteed to yield stable reduced systems, even if the target problem is stable. In the context of fluid flows, the skew-symmetric structure of the problem entails the preservation of the kinetic energy of the system. By preserving the same structure at the level of the reduced model, we obtain enhanced stability, and accuracy and the reduced model acquires physical significance by preserving a surrogate of the energy of the original problem. Next, we focus on Hamiltonian systems, which, being driven by symmetry, are a source of great interest in the reduction community. It is well known that the breaking of these symmetries in the reduced model is accompanied by a blowup of the system energy and flow volume. In this thesis, geometric reduced models for Hamiltonian systems are further developed and combined with the dynamically orthogonal methods, addressing the poor reducibility in time of advection-dominated problems. The reduced solution is expressed as a linear combination of a finite number of modes and coincides with the symplectic projection of the high-fidelity Hamiltonian problem onto the tangent space of the approximating manifold. An error surrogate is used to monitor the approximation ability of the reduced model and make a change in the rank of the approximating system if necessary. The method is further developed through a combination of DEIM and DMD to reduce non-polynomial nonlinearities while preserving the symplectic structure of the problem and applied to the Vlasov-Poisson system.In the second part of the thesis, we consider several data-driven methods to address the poor accuracy in the under-resolved regime for Galerkin reduced models via a closure term. The closure term is developed systematically from the Mori-Zwanzig formalism by introducing projection operators on the spaces of resolved and unresolved scales, thus resulting in an additional memory integral term. The interaction between different scales turns out to be nonlocal in time and dominated by a high-dimensional orthogonal dynamics equation, which cannot be solved precisely and efficiently. Several classical methods in the field of statistical mechanics are used to approximate the memory term, exploiting the finiteness of the memory kernel support. We conclude this thesis by showing through numerical experiments how long short-term memory networks, i.e., machine learning structures characterized by feedback connections, represent a valid tool for approximating the additional memory term.