Projection-based reduced order model and hyper-reduction of linear sloshing with geometric parameters using isogeometric analysis

Minimal subspace rotation on the Stiefel manifold for stabilization and enhancement of projection-based reduced order models for the compressible Navier–Stokes equations

In this contribution, we coupled the isogeometric analysis to a reduced order modelling technique in order to provide a computationally efficient solution in parametric domains. In details, we adopt the free-form deformation method to obtain the parametric formulation of the domain and proper orthogonal decomposition with interpolation for the computational reduction of the model. This technique provides a real-time solution for any parameter by combining several solutions, in this case computed using isogeometric analysis on different geometrical configurations of the domain, properly mapped into a reference configuration. We underline that this reduced order model requires only the full-order solutions, making this approach non-intrusive. We present in this work the results of the application of this methodology to a heat conduction problem inside a deformable collector pipe.

Many industrial chemical processes are complex, multi-phase and large scale in nature. These processes are characterized by various nonlinear physiochemical effects and fluid flows. Such processes often show coexistence of fast and slow dynamics during their time evolutions. The increasing demand for a flexible operation of a complex process, a pressing need to improve the product quality, an increasing energy cost and tightening environmental regulations make it rewarding to automate a large scale manufacturing process. Mathematical tools used for process modeling, simulation and control are useful to meet these challenges. Towards this purpose, development of process models, either from the first principles (conservation laws) i.e. the rigorous models or the input-output data based models constitute an important step. Both types of models have their own advantages and pitfalls. Rigorous process models can approximate the process behavior reasonably well. The ability to extrapolate the rigorous process models and the physical interpretation of their states make them more attractive for the automation purpose over the input-output data based identified models. Therefore, the use of rigorous process models and rigorous model based predictive control (R-MPC) for the purpose of online control and optimization of a process is very promising. However, due to several limitations e.g. slow computation speed and the high modeling efforts, it becomes difficult to employ the rigorous models in practise. This thesis work aims to develop a methodology which will result in smaller, less complex and computationally efficient process models from the rigorous process models which can be used in real time for online control and dynamic optimization of the industrial processes. Such methodology is commonly referred to as a methodology of Model (order) Reduction. Model order reduction aims at removing the model redundancy from the rigorous process models. The model order reduction methods that are investigated in this thesis, are applied to two benchmark examples, an industrial glass manufacturing process and a tubular reactor. The complex, nonlinear, multi-phase fluid flow that is observed in a glass manufacturing process offers multiple challenges to any model reduction technique. Often, the rigorous first principle models of these benchmark examples are implemented in a discretized form of partial differential equations and their solutions are computed using the Computational Fluid Dynamics (CFD) numerical tools. Although these models are reliable representations of the underlying process, computation of their dynamic solutions require a significant computation efforts in the form of CPU power and simulation time. The glass manufacturing process involves a large furnace whose walls wear out due to the high process temperature and aggressive nature of the molten glass. It is shown here that the wearing of a glass furnace walls result in change of flow patterns of the molten glass inside the furnace. Therefore it is also desired from the reduced order model to approximate the process behavior under the influence of changes in the process parameters. In this thesis the problem of change in flow patterns as result of changes in the geometric parameter is treated as a bifurcation phenomenon. Such bifurcations exhibited by the full order model are detected using a novel framework of reduced order models and hybrid detection mechanisms. The reduced order models are obtained using the methods explained in the subsequent paragraphs. The model reduction techniques investigated in this thesis are based on the concept of Proper Orthogonal Decompositions (POD) of the process measurements or the simulation data. The POD method of model reduction involves spectral decomposition of system solutions and results into arranging the spatio-temporal data in an order of increasing importance. The spectral decomposition results into spatial and temporal patterns. Spatial patterns are often known as POD basis while the temporal patterns are known as the POD modal coefficients. Dominant spatio-temporal patterns are then chosen to construct the most relevant lower dimensional subspace. The subsequent step involves a Galerkin projection of the governing equations of a full order first principle model on the resulting lower dimensional subspace. This thesis can be viewed as a contribution towards developing the databased nonlinear model reduction technique for large scale processes. The major contribution of this thesis is presented in the form of two novel identification based approaches to model order reduction. The methods proposed here are based on the state information of a full order model and result into linear and nonlinear reduced order models. Similar to the POD method explained in the previous paragraph, the first step of the proposed identification based methods involve spectral decomposition. The second step is different and does not involve the Galerkin projection of the equation residuals. Instead, the second step involves identification of reduced order models to approximate the evolution of POD modal coefficients. Towards this purpose, two different methods are presented. The first method involves identification of locally valid linear models to represent the dynamic behavior of the modal coefficients. Global behavior is then represented by ‘blending’ the local models. The second method involves direct identification of the nonlinear models to represent dynamic evolution of the model coefficients. In the first proposed model reduction method, the POD modal coefficients, are treated as outputs of an unknown reduced order model that is to be identified. Using the tools from the field of system identification, a blackbox reduced order model is then identified as a linear map between the plant inputs and the modal coefficients. Using this method, multiple local reduced LTI models corresponding to various working points of the process are identified. The working points cover the nonlinear operation range of the process which describes the global process behavior. These reduced LTI models are then blended into a single Reduced Order-Linear Parameter Varying (ROLPV) model. The weighted blending is based on nonlinear splines whose coefficients are estimated using the state information of the full order model. Along with the process nonlinearity, the nonlinearity arising due to the wear of the furnace wall is also approximated using the RO-LPV modeling framework. The second model reduction method that is proposed in this thesis allows approximation of a full order nonlinear model by various (linear or nonlinear) model structures. It is observed in this thesis, that, for certain class of full order models, the POD modal coefficients can be viewed as the states of the reduced order model. This knowledge is further used to approximate the dynamic behavior of the POD modal coefficients. In particular, reduced order nonlinear models in the form of tensorial (multi-variable polynomial) systems are identified. In the view of these nonlinear tensorial models, the stability and dissipativity of these models is investigated. During the identification of the reduced order models, the physical interpretation of the states of the full order rigorous model is preserved. Due to the smaller dimension and the reduced complexity, the reduced order models are computationally very efficient. The smaller computation time allows them to be used for online control and optimization of the process plant. The possibility of inferring reduced order models from the state information of a full order model alone i.e. the possibility to infer the reduced order models in the absence of access to the governing equations of a full order model (as observed for many commercial software packages) make the methods presented here attractive. The resulting reduced order models need further system theoretic analysis in order to estimate the model quality with respect to their usage in an online controller setting.

In this work, the saturation equation in the two phase flow model is transformed into quadratic bilinear form by introducing auxiliary variables. This new formulation is the first step for applying training-free reduced order modeling. There is no approximation involved in this transformation and it only increases the size of the problem linearly, before applying any model order reduction technique. Although this might seem counterintuitive to increase the size of the system, it can improve the basis selection and consequently the reduced system. Also, this formulation allows certain properties of the original model to be preserved in the reduced-order model, such as stability and passivity. A projection-based model order reduction on this form of system will yield a reduced-order model without the need to use anymore approximation such as discrete empirical interpolation method for evaluating nonlinear terms. Although in this work only the proper orthogonal decomposition is applied, this formulation is the first step for training free model order reduction after addressing the existing challenges. One of the main challenges is the dependency of the saturation equation on flux which is changing at different time steps, and thus makes the saturation equation time varying. The numerical results of applying the proposed formulation with POD model order reduction on a two-phase immiscible flow show a substantial reduction in the computational complexity.

Proper orthogonal decomposition with SUPG-stabilized isogeometric analysis for reduced order modelling of unsteady convection-dominated convection-diffusion-reaction problems

Model order reduction accelerated Monte Carlo stochastic isogeometric method for the analysis of structures with high-dimensional and independent material uncertainties

While machine learning algorithms have been around for a long time, many were developed under different names. Collectively, Projection-based Model Order Reduction (PMOR) methods constitute a case in point. They have been data-driven, physics-informed, machine learning methods since their inception, before these buzzwords became fashionable. For parametric linear problems, they are mature and have already made some impact in industry. For parametric nonlinear problems, their state of the art has been significantly advanced during the last decade on all of theoretical, algorithmic, and application fronts. Furthermore, many rational myth theories about PMOR have been debunked, particularly for highly nonlinear problems. Impressive results have been demonstrated for complex geometries and industrial-grade problems in many applications including nonlinear multi-scale solid mechanics, nonlinear structural dynamics with contact, convection-dominated turbulent flows, wave propagation in both the frequency and time domains, linear and nonlinear multi-disciplinary shape optimization, and uncertainty quantification. On the other hand, high-dimensional parameter spaces, topological changes, and the Kolmogorov n-width issue remain outstanding challenges; nevertheless, significant progress has also been made in these areas. First, this talk will rapidly overview the aforementioned state assessment of PMOR. Next, it will contrast the state of the art of this computational technology with alternative, popular, surrogate modeling techniques including those based on artificial neural networks, before presenting a recently developed, disruptive PMOR approach and reporting on its performance for real-world applications. Then, the talk will discuss the concept of mechanics-informed neural networks for data-driven constitutive modeling, where the informing process is not associated with any physics-based residual or differential equation, but with teaching a neural network mechanics concepts such as objectivity, isotropy (as needed), consistency, material stability, and dynamic stability. Finally, the talk will report on applications pertaining to the recent landing of Perseverance on Mars.

A data-driven model capable of predicting time-domain solutions of a high-fidelity tire–soil interaction model is developed to enable quick prediction of mobility capabilities on deformable terrain. The adaptive model order reduction based on the proper orthogonal decomposition (POD), for which the high-dimensional equations are projected onto the reduced subspace, is utilized as the basis for predicting the time-domain tire–soil interaction behavior. The projection-based model order reduction, however, requires many online matrix operations due to the successive updates of the nonlinear functions and Jacobians at every time-step, thereby hindering the computational improvement. Therefore, a data-driven approach using a long short-term memory (LSTM) neural network is introduced to predict the reduced order coordinates without the projection and time integration processes for computational speedup. With this model, a hybrid data-driven/physics-based off-road mobility model is proposed, where four separate LSTM-POD data-driven tire–soil interaction models are integrated into the physics-based multibody dynamics (MBD) vehicle model through a force–displacement coupling algorithm. By doing so, the individual data-driven tire–soil interaction model can be constructed efficiently, and the MBD and LSTM models are assembled as a single off-road mobility model and analyzed with existing off-road mobility solvers. The predictive ability and computational benefit of the proposed data-driven tire–soil interaction model with the POD-based model order reduction are examined with several numerical examples.

This work focuses on the development of accurate and fast simulation models for Ultra-Low Permeability (ULP) reservoirs, i.e., tight-sands and shales. ULP plays are the main unconventional resources that concentrate the bulk of production activity in the US. ULP challenge conventional simulators because they require multiphysics couplings, e.g., flow and geomechanics couplings, which poses a severe burden regarding computational efforts. We, thus address these challenges by developing accurate reduced-order models for coupled flow and geomechanics. We rely on projection-based Model-Order Reduction (MOR) and hyper-reduction (POD-DEIM) techniques to reduce the ULPs computational cost. More specifically, we perform the standard offline training stage on displacements as primary variables to create local basis using Proper Orthogonal Decomposition (POD). During the online phase, we project the residual and Jacobian that arise from both poroelasticity and rate-independent poroplasticity into the given basis to reduce one-way coupled flow and geomechanics computations. We approximate the tensors, for the energy equation, to minimize the serial-time. We consider the role of the heterogeneity and material models such as Von Mises and investigate the benefits of hyper-reduction (POD-DEIM) on the non-linear functions. Preliminary results, that focus on linear and nonlinear thermo-poroelasticity, show that our MOR algorithm provides substantial single and double digits speedups, up to 50X if we combine with multi-threading assembling and perform MOR on both physics. We highlight the remarkable MOR compression ratio above 99.9% for mechanics. The approach is particularly useful to speed up solving the sparse system for the inner iteration in convolution like problems which produces significant time savings compared to the serial full-order model (FOM). The latter is also true for problems that exhibit long serial times, for instance, while assembling the Jacobian and Residual for both physics and post-processing to compute stresses, if the serial-time per iteration is shorter that solving the sparse system of equations. These MOR results are promising in the sense that for most coupled flow and mechanics problems, the above condition holds. We formally compare FOM and reduced-order model (ROM) and provide time data to demonstrate the speedup of the procedure. Examples cover elasticity and rate-independent plasticity one-way coupled with the two-phase flow and the energy equation. We employ continuous Galerkin finite elements for the mechanics.

Simulating forced time-periodic flows in industrial applications presents significant computational challenges, partly due to the need to overcome costly transients before achieving time-periodicity. Reduced-order modelling emerges as a promising method to speed-up computations. We extend upon the work of Lotz et al. (2024) where a time-periodic space–time model is introduced. We present a time-periodic reduced-order model that directly finds the time-periodic solution without requiring extensive time integration. The reduced-order model gives a reduction in variables in both space and time. Our approach involves a POD-Galerkin reduced-order model based on a time-periodic full-order model that employs isogeometric analysis, residual-based variational multiscale turbulence modelling and weak boundary conditions. The projection-based reduced-order model inherits these features. We evaluate the reduced-order model with numerical experiments on moving hydrofoils. The motion is known a priori and we restrict ourselves to two spatial dimensions. In these experiments we vary the Strouhal and Reynolds numbers, and the motion profile respectively. Reduced-order model solutions agree well with those of the full-order model. The errors over the entire time period of thrust and lift forces are less than 0.2%. This includes complex scenarios such as the transition from drag to thrust production with increasing Strouhal number. Our time-periodic reduced-order model offers speed-ups ranging from O(102) to O(103) compared to the full-order model, depending upon the basis size. This makes it an appealing solution for prescribed time-periodic problems, with potential for additional speedup through nonlinear reduction techniques such as hyper-reduction.

Model order reduction approach generates lower dimensional approximations to the original system while preserving model’s essential information and computational accuracy. For nonlinear structural dynamic problems, where the stiffness matrix is configuration dependent, an iterative solution procedure is inevitable and a revisit to all the elements is essential for updating the stiffness matrix. In this paper, the nonlinear dynamics of the planar curved beams and 3D cylindrical shells are studied based on the isogeometric analysis and their model order reductions are investigated based on the proper orthogonal decomposition and discrete empirical interpolation method (POD-DEIM). Numerical results show that IGA-based POD-DEIM method significantly improves the computational efficiency of the nonlinear dynamic analysis of the beam and shell structures.

The Finite Element Method is a widely used discretization method for mechanical systems. Model Order Reduction is often applied to balance the need for accurate simulations with the requirement for acceptable simulation times by reducing the mathematical complexity of the system. One possible projection based Model Order Reduction method for linear systems is moment matching based on Krylov subspaces. In this method, the size of the reduced order model is directly proportional to the number of inputs and outputs of the system. Therefore, tangential directions can be applied to reduce the number of inputs used for Model Order Reduction. In this contribution, we examine the suitability of the Singular Value Decomposition Model Order Reduction (SVDMOR) method for linear elastic bodies in mechanical systems. SVDMOR is commonly used for electrical circuit simulation and reduces the number of inputs and outputs based on a singular value decomposition of the transfer function. The behavior of the singular values, which correspond to the error between the full order model and the model with a reduced number of inputs and outputs, as well as the tangential directions are investigated in the frequency domain on a numerical example.

Time-domain impedance boundary conditions for acoustic reduced order finite element simulations

ABSTRACTThe aeroelastic phenomenon of limit-cycle oscillations (LCOs) is analysed using a projection-based reduced-order model (PROM) and Navier–Stokes computational fluid dynamics (CFD) in the time domain. The proposed approach employs incompressible Navier–Stokes CFD to construct the full-order model flow field. A proper orthogonal decomposition (POD) of the snapshot matrix is conducted to extract the POD modes and corresponding temporal coefficients. The POD modes are directly projected to the incompressible Navier–Stokes equation to reconstruct the flow field efficiently. The methodology is applied to a plunging cylinder and an aerofoil undergoing LCOs. This scheme decreases the computational time while preserving the capability to predict the flow field accurately. The ROM is capable of reducing the computational time by at least 70% while maintaining the discrepancy within 0.1%. The causes of LCOs are also investigated. The scheme can be used to analyse non-linear aeroelastic phenomena in the time domain with reduced computational time.

We present a parametric reduced-order model for the neutral particle radiation transport equation. The approach devised is a minimally-intrusive, projection-based reduced-order model using global modes obtained via Proper Orthogonal Decomposition. The reduced-order model is specifically designed to work in a matrix-free fashion with radiation transport solvers relying on transport sweeps. The advantages and disadvantages of this model-order reduction approach are discussed and tested on two fixed-source radiation-transport benchmark problems, as well as a k-eigenvalue benchmark. The performance of the reduced-order model in terms of speedup and accuracy are found to be problem-dependent with speedup factors of around 2 for the fixed-source benchmarks and 43 for the k-eigenvalue benchmark. The corresponding reduced solution relative error for these speedups is 1% for the fixed-source benchmarks and 4 pcm for the k-eigenvalue benchmark.