Solvability Of System Research Articles

Thermo-economic investigation of transcritical Carbon Dioxide solar-thermal power plants, Part 1: System modeling and simulation for design with validation of results

This study comprises the engineering design and economic assessment of a novel trans-critical-point Carbon Dioxide cycle integrated with a solar Parabolic Trough Collector (PTC) power plant. A 1-MW facility with a Rankine transcritical CO2 cycle providing combined heat and power for a small industrial facility in Texas was proposed as the baseline power plant to perform the analysis. A design and modeling effort was performed in Engineering Equation Solver (EES) for the many systems and components of the power plant, including the solar field, the power block, and sub-components such as turbomachinery and heat exchangers. The model created enabled the simulation of plant performance for the median day of each month in the year by inputting meteorological TMY3 data, and results were extrapolated to estimate the annual energy and monetary income yield. Data processing was performed with MATLAB. Economic analysis included estimation of capital and O&M costs for the baseline plant. Results were verified through empirical performance simulation, using System Advisor Model (SAM), a software provided by NREL. The levelized cost of energy (LCOE) for the modeled plant was $0.2915/kWh, as calculated with EES/MATLAB, and $0.305/kWh, as calculated with SAM, resulting in a difference of less than 5%, between the two different simulation approaches. The design and simulation work presented herein provides the means to perform an optimization analysis, which reveals important clues as to how the energy cost could be reduced so as to make this technology a competitive alternative within the energy market.

Barrier-Augmented Lagrangian for GPU-based Elastodynamic Contact

We propose a GPU-based iterative method for accelerated elastodynamic simulation with the log-barrier-based contact model. While Newton's method is a conventional choice for solving the interior-point system, the presence of ill-conditioned log barriers often necessitates a direct solution at each linearized substep and costs substantial storage and computational overhead. Moreover, constraint sets that vary in each iteration present additional challenges in algorithm convergence. Our method employs a novel barrier-augmented Lagrangian method to improve system conditioning and solver efficiency by adaptively updating an augmentation constraint sets. This enables the utilization of a scalable, inexact Newton-PCG solver with sparse GPU storage, eliminating the need for direct factorization. We further enhance PCG convergence speed with a domain-decomposed warm start strategy based on an eigenvalue spectrum approximated through our in-time assembly. Demonstrating significant scalability improvements, our method makes simulations previously impractical on 128 GB of CPU memory feasible with only 8 GB of GPU memory and orders-of-magnitude faster. Additionally, our method adeptly handles stiff problems, surpassing the capabilities of existing GPU-based interior-point methods. Our results, validated across various complex collision scenarios involving intricate geometries and large deformations, highlight the exceptional performance of our approach.

Randomized methods for computing joint eigenvalues, with applications to multiparameter eigenvalue problems and root finding

AbstractIt is well known that a family of $${n}\times {n}$$ n × n commuting matrices can be simultaneously triangularized by a unitary similarity transformation. The diagonal entries of the triangular matrices define the n joint eigenvalues of the family. In this work, we consider the task of numerically computing approximations to such joint eigenvalues for a family of (nearly) commuting matrices. This task arises, for example, in solvers for multiparameter eigenvalue problems and systems of multivariate polynomials, which are our main motivations. We propose and analyze a simple approach that computes eigenvalues as one-sided or two-sided Rayleigh quotients from eigenvectors of a random linear combination of the matrices in the family. We provide some analysis and numerous numerical examples, showing that such randomized approaches can compute semisimple joint eigenvalues accurately and lead to improved performance of existing solvers.

Grassmann time-evolving matrix product operators: An efficient numerical approach for fermionic path integral simulations.

Developing numerical exact solvers for open quantum systems is a challenging task due to the non-perturbative and non-Markovian nature when coupling to structured environments. The Feynman-Vernon influence functional approach is a powerful analytical tool to study the dynamics of open quantum systems. Numerical treatments of the influence functional including the quasi-adiabatic propagator technique and the tensor-network-based time-evolving matrix product operator method have proven to be efficient in studying open quantum systems with bosonic environments. However, the numerical implementation of the fermionic path integral suffers from the Grassmann algebra involved. In this work, we present a detailed introduction to the Grassmann time-evolving matrix product operator method for fermionic open quantum systems. In particular, we introduce the concepts of Grassmann tensor, signed matrix product operator, and Grassmann matrix product state to handle the Grassmann path integral. Using the single-orbital Anderson impurity model as an example, we review the numerical benchmarks for structured fermionic environments for real-time nonequilibrium dynamics, real-time and imaginary-time equilibration dynamics, and its application as an impurity solver. These benchmarks show that our method is a robust and promising numerical approach to study strong coupling physics and non-Markovian dynamics. It can also serve as an alternative impurity solver to study strongly correlated quantum matter with dynamical mean-field theory.

Symbolic-numeric algorithm for parameter estimation in discrete-time models with exp

Dynamic models describe phenomena across scientific disciplines, yet to make these models useful in application the unknown parameter values of the models must be determined. Discrete-time dynamic models are widely used to model biological processes, but it is often difficult to determine these parameters. In this paper, we propose a symbolic-numeric approach for parameter estimation in discrete-time models that involve univariate non-algebraic (locally) analytic functions such as exp. We illustrate the performance (precision) of our approach by applying our approach to two archetypal discrete-time models in biology (the flour beetle ‘LPA’ model and discrete Lotka-Volterra competition model). Unlike optimization-based methods, our algorithm guarantees to find all solutions of the parameter values up to a specified precision given time-series data for the measured variables provided that there are finitely many parameter values that fit the data and that the used polynomial system solver can find all roots of the associated polynomial system with interval coefficients.

A theoretical and kinetic study of key reactions between ammonia and fuel molecules, part III: H-atom abstraction from esters by ṄH2 radicals

Hydrogen atom abstraction reactions by ṄH2 radicals play a crucial role in determining the reactivity of ammonia/fuel binary blends. Esters are a typical component of environmentally friendly and economically promising biofuels. The feasibility of the ammonia/biofuel dual-fuel approach has been proven in practical engines. [Energy and Fuels 22 (2008) 2963] and [Int. J. Energy Res. 2023 (2023) 9920670]. ṄH2 radicals play a critical role in the combustion and pyrolysis chemistry of ammonia and N-containing-rich fuels. In ammonia/biofuels hybrid combustion, ṄH2 radicals can react with biofuel molecules in a reaction class that is particularly important especially when sufficient ammonia is blended in order to eliminate NOx emissions. To help unravel the chemistry of ammonia/biofuel blends, a systematic theoretical kinetic study of H-atom abstraction from eleven alky esters of CnH2n+1COOCH3 (n = 1–4), CH3COOCmH2m+1 (m = 1–4), and C2H5COOC2H5, by ṄH2 radicals is performed in this work. The geometry optimization, frequency, and zero-point energy calculations for all related species, as well as the hindrance potential energy surface for low frequency torsional modes in the reactants and transition states, were performed at the M06–2X/6–311++G(d,p) level of theory. Intrinsic reaction coordinate calculations were performed to validate the connections between the transition states and expected minima energy species. The energies of all of the species involved were calculated at the QCISD(T)/cc-pVXZ (X = D, T, Q) and MP2/cc-pVYZ (Y = T, Q) levels of theory and then extrapolated to the complete basis set. Rate constants of 39 reactions were calculated using the Master Equation System Solver (MESS) program in the temperature range of 500 – 2000 K. These rate constants for different H-atom abstraction sites are provided and can be extrapolated to larger esters. The kinetic effects from the functional group are also illustrated by performing detailed comparisons with the previous studies of ṄH2 radical reactions with alkanes, alcohols and ethers.

PhotoSolver: A bidirectional photonic solver for systems of linear equations

Solving systems of linear equations is one of the most fundamental challenges in computing, with ubiquitous applications in science and engineering. Solving large-scale linear equations is particularly challenging, requiring tremendous computing expenses. However, conventional solvers in electronic digital computers face inevitable physical bottlenecks that hinder further advancements in electronic chips. Unlike the electronic architectures, optical computing harnesses the inherent advantages of lights, such as ultimately high speed, negligible energy consumption, and high parallelism, making optical architectures attractive sources for ultra-high speed analog computing. Especially, recent advances in integrated photonic circuits bring exciting new possibilities for high performance computing based on photonic chips. In this work, we propose a bidirectional PhotoSolver using the propagation of optical signals in different directions on a single integrated photonic chip to perform multiple operations in solving systems of linear equations, scalable to solve large-scale systems. Specifically, we propose an in situ approach to iteratively update solutions by performing the forward and backward propagations within a single photonic chip. Furthermore, we propose a partitioning approach to solve large-scale systems using arrays of photonic chips. Numerical experiments demonstrate these capabilities with representative systems of linear equations. By utilizing lights propagating in integrated photonic chips, our PhotoSolvers provide a promising computing architecture to solve large systems of linear equations, with potentials in wide computational applications.

A symmetric multigrid-preconditioned Krylov subspace solver for Stokes equations

Numerical solution of discrete PDEs corresponding to saddle point problems is highly relevant to physical systems such as Stokes flow. However, scaling up numerical solvers for such systems is often met with challenges in efficiency and convergence. Multigrid is an approach with excellent applicability to elliptic problems such as the Stokes equations, and can be a solution to such challenges of scalability and efficiency. The degree of success of such methods, however, is highly contingent on the design of key components of the multigrid scheme, including the hierarchy of discretizations, and the relaxation scheme used. Additionally, in many practical cases, it may be more effective to use a multigrid scheme as a preconditioner to an iterative Krylov subspace solver, as opposed to striving for maximum efficacy of the relaxation scheme in all foreseeable settings. In this paper, we propose an efficient symmetric multigrid preconditioner for the Stokes Equations on a staggered finite difference discretization. Our contribution is focused on crafting a preconditioner that (a) is symmetric indefinite, matching the property of the Stokes system itself, (b) is appropriate for preconditioning the SQMR iterative scheme [1], and (c) has the requisite symmetry properties to be used in this context. In addition, our design is efficient in terms of computational cost and facilitates scaling to large domains.

Unconditionally convergent [formula omitted] splitting iterative methods for variable coefficient Riesz space fractional diffusion equations

In this paper, we consider fast solvers for discrete linear systems generated by Riesz space fractional diffusion equations. We extract a scalar matrix, a compensation matrix, and a τ matrix from the coefficient matrix, and use their sum to construct a class of τ splitting iterative methods. Additionally, we design a preconditioner for the conjugate gradient method. Theoretical analyses show that the proposed τ splitting iterative methods are unconditionally convergent with convergence rates independent of step-sizes. Numerical results are provided to demonstrate the effectiveness of the proposed iterative methods.

A Note on the Convergence of Multigrid Methods for the Riesz–Space Equation and an Application to Image Deblurring

In recent decades, a remarkable amount of research has been carried out regarding fast solvers for large linear systems resulting from various discretizations of fractional differential equations (FDEs). In the current work, we focus on multigrid methods for a Riesz–Space FDE whose theoretical convergence analysis of such multigrid methods is currently limited in the relevant literature to the two-grid method. Here we provide a detailed theoretical convergence study in the multilevel setting. Moreover, we discuss its use combined with a band approximation and we compare the result with both τ and circulant preconditionings. The numerical tests include 2D problems as well as the extension to the case of a Riesz–FDE with variable coefficients. Finally, we investigate the use of a Riesz–Space FDE in a variational model for image deblurring, comparing the performance of specific preconditioning strategies.

A nonsmooth primal-dual method with interwoven PDE constraint solver

We introduce an efficient first-order primal-dual method for the solution of nonsmooth PDE-constrained optimization problems. We achieve this efficiency through not solving the PDE or its linearisation on each iteration of the optimization method. Instead, we run the method interwoven with a simple conventional linear system solver (Jacobi, Gauss–Seidel, conjugate gradients), always taking only one step of the linear system solver for each step of the optimization method. The control parameter is updated on each iteration as determined by the optimization method. We prove linear convergence under a second-order growth condition, and numerically demonstrate the performance on a variety of PDEs related to inverse problems involving boundary measurements.

Stabilization of SIMPLE-like RANS solvers for computing accurate gradients using the complex-step derivative method

The Reynolds-averaged Navier-Stokes (RANS) approach in Computational Fluid Dynamics (CFD) is increasingly vital for aerodynamic design across wind turbines, gas turbines, aircraft, and rotorcraft. Enhancing the process through CFD-based design optimization requires numerous design variables, favoring gradient-based methods for better scalability than gradient-free methods. This research uses numerical differentiation techniques, particularly the complex-step derivative method, to compute gradients. Challenges arise during aerodynamic shape optimization when unconventional shapes disrupt RANS solver assumptions, causing convergence failures. These failures undermine optimization, prompting the need to enhance solver convergence for robust optimization. The modified-Boostconv method is a residual recombination method bolstering unstable eigenvalues to stabilize convergence of iterative solvers for nonlinear systems of equations. This study extends the modified-Boostconv method to combine it with the complex-step derivative technique, creating robust optimizations via RANS-CFD with accurate gradients, even for these numerically unstable cases. The main issue encountered during the complexification of the modified-Boostconv method is how to correctly ensure that the model reduction leading to the least-squares problem satisfies the complex-step derivative method. We test whether the dot product operation involved in the model reduction should be a Hermitian or non-Hermitian inner product. The problem is first tested for a simple analytical case using the logistic equation in combination with the modified-Boostconv method. It shows that the non-Hermitian inner product should be used. This is also confirmed with a similar study using the RANS solver.

Recycling Krylov subspaces for efficient partitioned solution of aerostructural adjoint systems

Robust and efficient solvers for coupled-adjoint linear systems are crucial to successful aerostructural optimization. Monolithic and partitioned strategies can be applied. The monolithic approach is expected to offer better robustness and efficiency for strong fluid-structure interactions. However, it requires a higher implementation cost and convergence may depend on appropriate scaling and initialization strategies. On the other hand, the modularity of the partitioned method enables a straightforward implementation while its convergence may require relaxation. In addition, a partitioned solver often leads to a higher number of iterations to get the same level of convergence as the monolithic one.The objective of this paper is to accelerate the fluid-structure coupled-adjoint partitioned solver by considering techniques borrowed from approximate invariant subspace recycling strategies adapted to sequences of linear systems with varying right-hand sides. Indeed, in a partitioned framework, the structural source term attached to the fluid block of equations affects the right-hand side with the nice property of quickly converging to a constant value. For the fluid block of equations, we also consider deflation of approximate eigenvectors in conjunction with advanced inner-outer Krylov solvers.To demonstrate the effectiveness of our proposed recycling strategy, we compute the coupled derivatives for two emblematic problems in transonic viscous flow. The first one is an aeroelastic configuration of the ONERA-M6 fixed wing. For this exercise, the fluid grid was coupled to a structural model specifically designed to exhibit a high flexibility. The second one is a wing-body aeroelastic configuration of the Common Research Model. All computations are performed using a fully linearized one-equation Spalart-Allmaras turbulence model. Numerical simulations show up to 39% reduction in matrix-vector products for GCRO-DR and up to 25% for the nested FGCRO-DR solver.

A New Optimal Numerical Root-Solver for Solving Systems of Nonlinear Equations Using Local, Semi-Local, and Stability Analysis

This paper introduces an iterative method with a remarkable level of accuracy, namely fourth-order convergence. The method is specifically tailored to meet the optimality condition under the Kung–Traub conjecture by linear combination. This method, with an efficiency index of approximately 1.5874, employs a blend of localized and semi-localized analysis to improve both efficiency and convergence. This study aims to investigate semi-local convergence, dynamical analysis to assess stability and convergence rate, and the use of the proposed solver for systems of nonlinear equations. The results underscore the potential of the proposed method for several applications in polynomiography and other areas of mathematical research. The improved performance of the proposed optimal method is demonstrated with mathematical models taken from many domains, such as physics, mechanics, chemistry, and combustion, to name a few.

Towards a machine-learned Poisson solver for low-temperature plasma simulations in complex geometries

Poisson’s equation plays an important role in modeling many physical systems. In electrostatic self-consistent low-temperature plasma (LTP) simulations, Poisson’s equation is solved at each simulation time step, which can amount to a significant computational cost for the entire simulation. In this paper, we describe the development of a generic machine-learned Poisson solver specifically designed for the requirements of LTP simulations in complex 2D reactor geometries on structured Cartesian grids. Here, the reactor geometries can consist of inner electrodes and dielectric materials as often found in LTP simulations. The approach leverages a hybrid CNN-transformer network architecture in combination with a weighted multiterm loss function. We train the network using highly randomized synthetic data to ensure the generalizability of the learned solver to unseen reactor geometries. The results demonstrate that the learned solver is able to produce quantitatively and qualitatively accurate solutions. Furthermore, it generalizes well on new reactor geometries such as reference geometries found in the literature. To increase the numerical accuracy of the solutions required in LTP simulations, we employ a conventional iterative solver to refine the raw predictions, especially to recover the high-frequency features not resolved by the initial prediction. With this, the proposed learned Poisson solver provides the required accuracy and is potentially faster than a pure GPU-based conventional iterative solver. This opens up new possibilities for developing a generic and high-performing learned Poisson solver for LTP systems in complex geometries.

An Optimal Linear-combination-of-unitaries-based Quantum Linear System Solver

Solving systems of linear equations is one of the most important primitives in many different areas, including in optimization, simulation, and machine learning. Quantum algorithms for solving linear systems have the potential to provide a quantum advantage for these problems. In this work, we recall the Chebyshev iterative method and the corresponding optimal polynomial approximation of the inverse. We show that the Chebyshev iteration polynomial can be efficiently evaluated both using quantum singular value transformation (QSVT) as well as linear combination of unitaries (LCU). We achieve this by bounding the 1-norm of the coefficients of the polynomial expressed in the Chebyshev basis. This leads to a considerable constant-factor improvement in the runtime of quantum linear system solvers that are based on LCU or QSVT (or, conversely, a several orders of magnitude smaller error with the same runtime/circuit depth).

Index‐aware learning of circuits

AbstractElectrical circuits are present in a variety of technologies, making their design an important part of computer aided engineering. The growing number of parameters that affect the final design leads to a need for new approaches to quantify their impact. Machine learning may play a key role in this regard; however, current approaches often make suboptimal use of existing knowledge about the system at hand. In terms of circuits, their description via modified nodal analysis is well‐understood. This particular formulation leads to systems of differential‐algebraic equations (DAEs), which bring with them a number of peculiarities, for example, hidden constraints that the solution needs to fulfill. We use the recently introduced dissection index that can decouple a given system of DAEs into ordinary differential equations, only depending on differential variables, and purely algebraic equations, that describe the relations between differential and algebraic variables. The idea is to then only learn the differential variables and reconstruct the algebraic ones using the relations from the decoupling. This approach guarantees that the algebraic constraints are fulfilled up to the accuracy of the nonlinear system solver, and it may also reduce the learning effort as only the differential variables need to be learned.

Development and validation of a new one-dimensional parallel transient system solver with coordinate transformation method developed in OpenFOAM

Passive containment cooling system is an important component of both 3rd and 4th generation nuclear power plants. To simulate two-phase flow in loop-type passive containment cooling system, a new one-dimensional parallel transient system solver, ComspaFOAM-SYS1D, was developed based on OpenFOAM. Compared to the previously developed one-dimensional solver, the new solver can be accelerated using multithreading. Additionally, a new coordinate transformation method was used in ComspaFOAM-SYS1D, enabling OpenFOAM to simulate a one-dimensional loop with bends. Faucet problem was used to verify the basic equations and models in the new solver. Validation was undertaken based on the GENEVA facility and the passive containment cooling system comprehensive performance test facility. In the assessments, representative parameters such as void fraction, pressure, temperature, phase velocity, and oscillation period were compared. Regarding efficiency, the simulation results indicated that the parallel speedup of the new solver could approach a value of approximately 2.0. In terms of accuracy, the relative deviations of time-averaged and transient extremum values of all the simulation results from the experimental results were within 15.0%.

Parallel multi‐rate simulation scheme for modular multilevel converter‐based high‐voltage direct current with accurate simulation of high‐frequency characteristics and field programmable gate array‐based implementation

AbstractThe real‐time simulation of the modular multilevel converter‐based high‐voltage direct current (MMC‐HVDC) transmission system has become a popular research topic. However, in order to meet the real‐time performance, the real‐time simulation technology will cause additional simulation errors for MMC‐HVDC, especially on its frequency characteristics. Therefore, a parallel multi‐rate simulation scheme for MMC‐HVDC is developed in this work to ensure accurate simulation of high‐frequency characteristics. Firstly, a non‐error method based on converter transformer decoupling is proposed to decouple the converter and alternating current system; direct current transmission line decoupling and arm decoupling methods are used to achieve decoupling among and within converters. A multi‐rate data synchronous mechanism is established by considering the differences among high‐frequency characteristics caused by delayed data interaction. Secondly, the computing architectures of the primary system solver and modular multilevel converter controller are designed based on a field programmable gate array (FPGA). The real‐time simulation platform for a four‐terminal true bipolar MMC‐HVDC is constructed based on the FPGA array. Thirdly, the factors in multi‐rate simulation affecting the simulation accuracy of high‐frequency characteristics are analysed. The simulator is shown to be accurate in steady and dynamic states. The authors also verify its applicability for further research on high‐frequency resonance based on control experiments.

An error predict-correction formula of the load vector in the BSLM for solving three-dimensional Burgers’ equations

This paper aims to develop an algorithm reducing the computational cost of the backward semi-Lagrangian method for solving nonlinear convection–diffusion equations. For this goal, we introduce an error predict-correction formula (EPCF) of the load term for the Helmholtz system. The EPCF is built up to involve the same values as solving the perturbed Cauchy problem, which allows the reuse of the values. These reuses dramatically reduce the computational cost by eliminating the interpolation processes to evaluate the load term in most computational domains. The algorithm also produces a stable solution, even where the original solution has a sharp gradient, by introducing an indicator to distinguish the sharpness of the solution. Furthermore, the proposed algorithm discusses fast solvers of the discrete systems for the perturbed Cauchy problem, for which we utilize an eigenvalue decomposition theory. To show the adaptability and efficiency of the proposed algorithm, we apply it to the convection–diffusion equations such as two and three-dimensional viscous Burgers’ equations. Through several simulations, we confirm that the proposed algorithm for the load term radically reduces the computational cost, and the indicator for stiff components guarantees precision.