Fast Numerical Approximation of Parabolic Problems Using Model Order Reduction and the Laplace Transform

During lamination, residual thermo-mechanical stresses are induced in the encapsulated solar cells composing photovoltaic (PV) modules. Depending on the material and geometrical configuration of the layers of the laminate, this residual stress field can be beneficial since it may lead to a compressive stress state in Silicon and therefore crack closure effects in the presence of cracks, with a recovery of electrical conductivity in cracked solar cells. It is therefore important to investigate the distribution of thermo-mechanical stresses within the PV laminate with a view to optimizing the coupling between the electrical response and elastic deformation in the operation of PV modules. A promising approach proposed in the present thesis regards the prediction of residual stresses in composite laminates by using a shear-lag theory to model the epoxy-vinil-acetate polymeric layers, accounting for their thermo-visco-elastic response. Moreover, it will be shown that thermomechanical formulations for stress analysis of a PV laminate lead to a system of higher order ordinary differential equations or partial differential equations in which the exact solutions may be impossible to be determined in closed form and hence numerical schemes become desirable. However, the computational cost associated with the implementation of the numerical scheme may be significantly expensive. Therefore, a method to reduce the computational complexity is expected to be very important. To this aim, Model Order Reduction (MOR) techniques are applied hierarchically, first to the thermal system of a PV module in service, and then extended to coupled thermo-mechanical problems. A combination of proper orthogonal decomposition (POD) and discrete empirical interpolation method (DEIM) with a modified formulation is proposed for the first-order thermal equations of photovoltaic system during service and a new coupled second-order Krylov based formulation is developed for model order reduction of the coupled thermo-mechanical model of the photovoltaic module. The results of these reduction schemes show a huge computational gain in the reduced system solutions and a high accuracy of the reduced system outputs

In this work, a model order reduction (MOR) technique for heat-transfer system-level modeling is presented. A heat transfer solver using the finite difference method (FDM), which is appropriate for MEMS thermal analysis, is implemented. The numerical models generated by the FDM solver then were successfully reduced into low-order compact models by an Arnoldi-based model-order-reduction technique. The compact macromodels can be easily inserted into system-level or circuit simulators. We have demonstrated that the macromodel results for a thermal actuator are in good agreement with the experimental results. We also use the same technique on periodic surface heat flux problems to observe the relationship between frequency and minimum required order of the reduced model under specific error tolerance.

In steam turbines (STs), evaluation technology of thermal deformations during transient operations is increasingly important, because of a demand for improvement of operability such as shortening startup time. However, it is still difficult to predict temperatures of metal parts with sufficient accuracy due to complexity of heat transfer phenomena. The present study puts its focus on developing a method to identify metal temperatures of STs during start-up operations using particle filter (PF), a kind of sequential Bayesian filter utilizing ensemble approximation, and model order reduction (MOR) technique. In this method, a self-organizing state-space model is used to calculate time evaluations of metal temperatures, and heat transfer coefficient (HTC) parameters and steam bulk temperature parameters. And an optimal estimate of these is obtained through iterations of a short-time prediction and a correction of variables using measured temperatures. In this state-space model, state variables including HTC and bulk temperature parameters are treated as random variables, and the time evolution of each variable is modeled as follows; (1) Metal temperature is modeled using a reduced order model (ROM) in order to reduce computational time of PF. The ROM is constructed from a finite element model for unsteady heat transfer analysis using MOR technique. (2) HTC is modeled as a random walk model where its changes during one time step is randomly determined using Gaussian noise. Here, the magnitude of this noise is adjusted automatically. (3) Bulk temperature is modeled using prediction formulas with Gaussian noise added. As a validation problem, a cold start-up operation of a ST unit of a gas turbine combined cycle (GTCC) is considered. The proposed method is applied to the problem, and boundary conditions (BCs) are identified using measured temperatures at 68 measurement points. Then, the heat transfer analysis based on finite element analysis (FEA) is performed using the identified BCs. As a result of this FEA, it is confirmed that the calculated metal temperature tends to agree better with measurements compared with that of initial FEA, and that the errors of the calculated temperature at measurement points reduce by 41% on average compared with initial FEA. From this result, it is concluded that this proposed method is effective for metal temperature identification during start-up operations.

Microgrids (MGs) concepts are a systematic way to integrate the supply and demand side along with the storage devices. These mostly use renewable energy resources (RESs) viz. solar PV and wind turbines, interfaced to the low voltage distribution network via an inverter. As several components are involved, MGs has a system of systems (SoSs) architecture and its modeling leads to form a high order system. Hence, for the controllers' design purposes, these models had to be brought down to a reduced lower order system. This paper performs a comparative study of various model order reduction (MOR) techniques viz. aggregation, balanced truncation (BT), singular perturbation (SP) and Hankel norm approximation (HNA) on a two-source autonomous MG system. The standard performance evaluation methods used for a multi-input multi-output (MIMO) system i.e. the step and impulse responses, eigenvalue analysis and Bode plots were considered to determine the effectiveness of these MOR method.

I consider the problem of computing the space of conservation laws for a second-order parabolic partial differential equation for one function of three independent variables. The PDE is formulated as an exterior differential system \({\cal I}\) on a 12-manifold M, and its conservation laws are identified with the vector space of closed 3-forms in the infinite prolongation of \({\cal I}\) modulo the so-called "trivial" conservation laws. I use the tools of exterior differential systems and Cartan's method of equivalence to study the structure of the space of conservation laws. My main result is: Theorem.Any conservation law for a second-order parabolic PDE for one function of three independent variables can be represented by a closed 3-form in the differential ideal ${\cal I}$ on the original 12-manifold M. I show that if a nontrivial conservation law exists, then \({\cal I}\) has a deprolongation to an equivalent system \({\cal J}\) on a 7-manifold N, and any conservation law for \({\cal I}\) can be expressed as a closed 3-form on N that lies in \({\cal J}\). Furthermore, any such system in the real analytic category is locally equivalent to a system generated by a (parabolic) equation of the formA (uxxuyy-u2xy)+Buxx+2Cuxy+Duyy+E = 0 where A, B, C, D, E are functions of x, y, t, u, ux, uy, ut. I compute the space of conservation laws for several examples, and I begin the process of analyzing the general case using Cartan's method of equivalence. I show that the non-linearizable equation \( u_t = {1 \over 2} e^{-u} (u_{xx}+u_{yy}) \) has an infinite-dimensional space of conservation laws. This stands in contrast to the two-variable case, for which Bryant and Griffiths showed that any equation whose space of conservation laws has dimension 4 or more is locally equivalent to a linear equation, i.e., is linearizable.

We show the continuous dependence of solutions of linear nonautonomous second-order parabolic partial differential equations (PDEs) with bounded delay on coefficients and delay. The assumptions are very weak: only convergence in the weak-* topology of delay coefficients is required. The results are important in the applications of the theory of Lyapunov exponents to the investigation of PDEs with delay.

Background This study establishes a mathematically consistent and computational framework for the simultaneous identification of two time-dependent coefficients in a one-dimensional second-order parabolic partial differential equation. The considered problem is governed by nonlocal initial, boundary, and integral overdetermination conditions. Methods The direct problem is solved using the Crank-Nicolson finite difference method (FDM), which ensures unconditional stability and second-order accuracy in both spatial and temporal discretizations. The corresponding inverse problem is reformulated as a nonlinear regularized least-squares optimization problem and efficiently solved used the MATLAB subroutine lsqnonlin from the optimization Toolbox. Due to the intrinsic, ill-posedness of the inverse formulation, small input data errors lead to big output errors. Then, Tikhonov regularization, is employed to enhance numerical stability and robustness. Results Extensive numerical experiments are carried out under exact and noisy data to evaluate the numerical accuracy and convergence behavior of the method. The results confirm that the regularization technique effectively damps numerical oscillations, minimizes reconstruction error, and ensures reliable recovery of the unknown coefficients. Sensitivity analysis further reveals the essential role of the regularization parameter in controlling the trade-off between stability and accuracy. Conclusions The proposed approach provides an accurate and computationally efficient tool for IP in heat transfer, diffusion processes, and related applied sciences.

The problem of calculating the evaporation from the soil surface is formulated as an optimal control problem. The controlled process of the vertical water transfer in soil is described by a onedimensional nonlinear second-order parabolic partial differential equation. The objective function is the squared Euclidean distance between the calculated values of the soil moisture at various depths and certain prescribed values. To improve the efficiency of finding a numerical solution, the sensitivity of the soil moisture at various depths to the variations of evaporation is estimated by means of fast automatic differentiation. The analysis of these estimates made it possible to determine an effective thin subsurface soil layer in which the moisture is most sensitive to the variations of evaporation; it is in this soil layer where the objective function should be calculated.

Second-order parabolic partial differential equations are solved by using a new three level method based on non-polynomial cubic spline in the space direction and finite difference in the time direction. Stability analysis of the method has been carried out and we have shown that our method is unconditionally stable. It has been shown that by suitably choosing the parameters most of the previous known methods for homogeneous and non-homogeneous cases can be obtained from our method. We also obtain a new high accuracy scheme of O(k 4+h 4). Numerical examples are given to illustrate the applicability and efficiency of the new method.

System of Nonlinear Second-Order Parabolic Partial Differential Equations with Interconnected Obstacles and Oblique Derivative Boundary Conditions on Non-Smooth Time-Dependent Domains

The negative environmental impact of carbon emissions from fossil fuels has promoted hydrogen utilization and storage in underground structures. Hydrogen leakage from storage structures through wells is a major concern due to the small hydrogen molecules that diffuse fast in the porous well cement sheath. The second-order parabolic partial differential equation describing the hydrogen diffusion in well cement was solved numerically using the finite difference method (FDM). The numerical model was verified with an analytical solution for an ideal case where the matrix and fluid have invariant properties. Sensitivity analyses with the model revealed several possibilities. Based on simulation studies and underlying assumptions such as non-dissolvable hydrogen gas in water present in the cement pore spaces, constant hydrogen diffusion coefficient, cement properties such as porosity and saturation, etc., hydrogen should take about 7.5 days to fully penetrate a 35 cm cement sheath under expected well conditions. The relatively short duration for hydrogen breakthrough in the cement sheath is mainly due to the small molecule size and high hydrogen diffusivity. If the hydrogen reaches a vertical channel behind the casing, a hydrogen leak from the well is soon expected. Also, the simulation result reveals that hydrogen migration along the axial direction of the cement column from a storage reservoir to the top of a 50 m caprock is likely to occur in 500 years. Hydrogen diffusion into cement sheaths increases with increased cement porosity and diffusion coefficient and decreases with water saturation (and increases with hydrogen saturation). Hence, cement with a low water-to-cement ratio to reduce water content and low cement porosity is desirable for completing hydrogen storage wells.

In this paper, we consider a class of optimal control problems with control and terminal inequality constraints, where the system dynamics is governed by a linear second-order parabolic partial differential equation with first boundary condition. A feasible direction algorithm for solving this class of optimal control problems has already been obtained in the literature. The aim of this paper is to improve the convergence result by using a topology arising in the study of relaxed controls.

A highly accurate algorithm for retrieving the predicted behavior of problems with piecewise-smooth initial data

Nonlinear Model Order Reduction for Catalytic Tubular Reactors

Coupled flow and geomechanics computations are very complex and require solving large nonlinear systems. Such simulations are intense from both runtime and memory standpoint, which strongly hints at employing model order reduction (MOR) techniques to speed them up. Different types of Reduced-Order Models (ROM) have been proposed to alleviate this computational burden. MOR approaches rely on projection operators to decrease the dimensionality of the problem. We first execute a computationally expensive “offline” stage, during which we carefully study the full order model (FOM). Upon creating a ROM basis, we then perform the cheap “online” stage. Our reduction strategy estimates a ROM using proper orthogonal decomposition (POD). We determine a family of solutions to the problem, for a suitable sample of input conditions, where every single realization is so-called a “snapshot.” We then ensemble all snapshots to determine a compressed subspace that spans the solution. Usually, POD employs a fixed reduced subspace of global basis vectors. The usage of a global basis is not convenient to tackle problems characterized by different physical regimes, parameter changes, or high-frequency features. Having many snapshots to capture all these variations is unfeasible, which suggests seeking adaptive approaches based on the closest regional basis. We thus develop such a strategy based on local POD basis to reduce one-way coupled flow and geomechanics computations. We partition the time window to adequately capture regimes such as depletion/build-up and decreasing the number of snapshots per basis. We focus on linear elasticity and consider factors such as the role of the heterogeneity. We also assess how to tackle different degrees of freedom, such as the displacements (intercalated and coupled), pressure, and temperature, with MOR. Preliminary 2- and 3-D results show significant compression ratios up to 99.9% for the mechanics part. We formally compare FOM and ROM and provide time data to demonstrate the speedup of the procedure. Examples focus on linear and nonlinear poroelasticity. We employ continuous Galerkin finite elements for all of the discretizations.