Numerical solutions for the Poisson-Nernst-Planck system using integrated radial basis function and moving least squares techniques with reduced order method

In this paper, an extreme value problem on the wave-induced vertical bending moment (WVBM) of a ship is addressed. The First Order Reliability Method (FORM) is adopted to predict the extreme value distribution of the WVBM under given short-term sea states. The one-way coupled Computational Fluid Dynamics (CFD) and Finite Element Analysis (FEA) is employed to obtain numerical solutions of the WVBM. To predict the extreme value distribution based on the coupled CFD-FEA, the Reduced Order Method (ROM) is further introduced as a surrogate model for the CFD-FEA. By combining the ROM and FORM, the extreme value prediction equivalent to the coupled CFD-FEA can be estimated in a robust but inexpensive manner. The present ROM is composed of the transfer function (TF) of the WVBM and a simple correction for the nonlinearity. The accuracy of the ROM is validated by comparing with the coupled CFD-FEA results. A series of numerical demonstrations on the extreme value distribution of the WVBM using the ROM and FORM under moderate and severe sea conditions is given.

Single- and multi-objective optimization of an aircraft hot-air anti-icing system based on Reduced Order Method

SummaryIn this work, we present an approach for the efficient treatment of parametrized geometries in the context of proper orthogonal decomposition (POD)‐Galerkin reduced order methods based on finite‐volume full order approximations. On the contrary to what is normally done in the framework of finite‐element reduced order methods, different geometries are not mapped to a common reference domain: the method relies on basis functions defined on an average deformed configuration and makes use of the discrete empirical interpolation method to handle together nonaffinity of the parametrization and nonlinearities. In the first numerical example, different mesh motion strategies, based on a Laplacian smoothing technique and on a radial basis function approach, are analyzed and compared on a heat transfer problem. Particular attention is devoted to the role of the nonorthogonal correction. In the second numerical example, the methodology is tested on a geometrically parametrized incompressible Navier‐Stokes problem. In this case, the reduced order model is constructed following the same segregated approach used at the full order level.

Reduced order unsteady aerodynamic modeling for stability and control analysis using computational fluid dynamics

In this article, we present a two‐level implicit difference scheme for Korteweg–de Vires equation with the initial and boundary conditions by the method of order reduction. The truncation error of the difference scheme is analyzed in detail. In the practical computation, the introduced intermediate variable is decoupled in order to reduce the computational cost. It is proved that the difference scheme is solvable by the Browder theorem. The conservation, boundedness, and the unconditional convergence of the numerical solution are also analyzed at length. The convergence order is two both in space and in time in L2‐norm. The numerical solution is proved to be unique under the optimal step ratio condition. Numerical examples demonstrate that the theoretical analysis is correct.

The method consists in integrating the Orr-Sommerfeld equation in the direction from the free stream toward the wall. In order to satisfy the boundary conditions at the wall, two linearly independent solutions have to be found. To prevent numerical solutions from becoming linearly dependent, the method of order reduction instead of repeated orthogonalization has been used. The method has been applied to calculate the neutral curve for the Blasius profile.

The modified phase field crystal model is a sixth order nonlinear generalized damped wave equation. Two linearized difference schemes are presented based on the method of order reduction. One scheme is second order both in time and space, and the other scheme is convergent with the second temporal order and fourth spatial order. A theoretical analysis is carried out by the energy argument and mathematical induction. The uniqueness of the numerical solution and unconditional convergence in discrete -norm are proved rigorously. Numerical results demonstrate that the presented schemes for the modified phase field crystal equation can achieve the theoretical convergence order.

In this paper, we develop a numerical method with a reduced order technique based on the Galerkin procedure to solve nonlinear partial differential equations of MEMS devices. We apply an explicit numerical approach based on the finite difference method (FDM) to a reduced order model of the equation of micro-beams and call it explicit-ROM. As a case study, we obtain the time response of a micro-beam under an electrostatic actuation and a mechanical shock with our method and the reduced order method (ROM) developed in previous papers. We show ROM requires taking the effect of higher order modes into consideration in order to result in accurate response, while explicit-ROM greatly improves both accuracy and speed with only the first mode, hence it is a straightforward approach that can be used in a MEMS software to obtain very fast and accurate results.

This work considers the use of Adam Bashforth-Moulton method and Milne Simpson method to solve second order ordinary differential equation with initial value problem and to compare solution with the exact solution, to solve that we first convert the equation to two set of first order differential equation by order reduction method, then we use a single step method for approximation of initially orders which are required to start the linear multistep method. The result show that the numerical solutions are in good agreement with the exact solution. The result show that Adam Bashforth-Moulton method is better than Milne Simpson method in solving differential equation of second order.

A nonlinear beam formulation is presented based on the Gurtin-Murdoch surface elasticity and the modified couple stress theory. The developed model theoretically takes into account coupled effects of the energy of surface layer and microstructures sizedependency. The mid-plane stretching of a beam is incorporated using von-Karman nonlinear strains. Hamilton’s principle is used to determine the nonlinear governing equation of motion and the corresponding boundary conditions. As a case study, pull-in instability of an electromechanical nano-bridge structure is studied using the proposed formulation. The nonlinear governing equation is solved by the analytical reduced order method (ROM) as well as the numerical solution. Effects of various parameters including surface layer, size dependency, dispersion forces, and structural damping on the pullin parameters of the nano-bridges are discussed. Comparison of the results with the literature reveals capability of the present model in demonstrating the impact of nanoscale phenomena on the pull-in threshold of the nano-bridges.

This paper introduces the application of the model order reduction (MOR) method using Laguerre polynomials in building envelope simulation. The dynamic heat transfer problems of a single-layer roof with the sinusoidal outdoor temperature as well as a multi-layer wall and a two-dimensional corner of wall under the realistic outside solar-air temperature were solved by the direct numerical solution method and Laguerre polynomial model order reduction (LP-MOR) method. The relative error, solution time, and heat flux of the LP-MOR and other MOR methods were compared. The maximum norm of relative errors between LP-MOR and the directly direct numerical solution (DNS) method based on finite volume method (FVM) can be controlled within . Moreover, compared with solving small-scale discrete systems, LP-MOR is more efficient in solving complex models with a large number of discrete nodes.

A greedy nonintrusive reduced order method (ROM) is proposed for parameterized time-dependent problems with an emphasis on problems in fluid dynamics. The nonintrusive ROM (NIROM) bases on a two-level proper orthogonal decomposition to extract temporal and spatial reduced basis from a set of candidates, and adopts the radial basis function to approximate undetermined coefficients of extracted reduced basis. Instead of adopting uniform or random sampling strategies, the candidates are determined by an adaptive greedy approach to minimize the overall offline computational cost. Numerical studies are presented for a two-dimensional diffusion problem as well as a lid-driven cavity problem governed by incompressible Navier–Stokes equations. The results demonstrate that the greedy nonintrusive ROM (GNIROM) predicts the flow field accurately and efficiently.

The neutronics and thermal-hydraulics (N/TH) coupling behavior analysis is a key issue for nuclear power plant design and safety analysis. Due to the high-dimensional partial differential equations (PDEs) derived from the N/TH system, it is usually time consuming to solve such a large-scale nonlinear equation by the traditional numerical solution method of PDEs. To solve this problem, this work develops a reduced order model based on the proper orthogonal decomposition (POD) and artificial neural networks (ANNs) to simulate the N/TH coupling system. In detail, the POD method is used to extract the POD modes and corresponding coefficients from a set of full-order model results under different boundary conditions. Then, the backpropagation neural network (BPNN) is utilized to map the relationship between the boundary conditions and POD coefficients. Therefore, the physical fields under the new boundary conditions could be calculated by the predicated POD coefficients from ANN and POD modes from snapshot. In order to assess the performance of an ANN-POD-based reduced order method, a simplified pressurized water reactor model under different inlet coolant temperatures and inlet coolant velocities is utilized. The results show that the new reduced order model can accurately predict the distribution of the physical fields, as well as the effective multiplication factor in the N/TH coupling nuclear system, whose relative errors are within 1%.

We consider the numerical solution of large-scale symmetric differential matrix Riccati equations. Under certain hypotheses on the data, reduced order methods have recently arisen as a promising cl...

To overcome the shortcomings of extreme time-consuming in solving the Reynolds equation, two efficient calculation methods, based on the free boundary theory and variational principles for the unsteady nonlinear Reynolds equation in the condition of Reynolds boundary, are presented in the paper. By employing the two mentioned methods, the nonlinear dynamic forces as well as their Jacobians of the journal bearing can be calculated saving time but with the same accuracy. Of these two methods, the one is called a Ritz model which manipulates the cavitation region by simply introducing a parameter to match the free boundary condition and, as a result, a very simple approximate formulae of oil-film pressure is being obtained. The other one is a one-dimensional FEM method which reduces the two-dimensional variational inequality to the one-dimensional algebraic complementary equations, and then a direct method is being used to solve these complementary equations, without the need of iterations, and the free boundary condition can be automatically satisfied. Meanwhile, a new order reduction method is contributed to reduce the degrees of freedom of a complex rotor-bearing system. Thus the nonlinear behavior analysis of the rotor-bearing system can be studied time-sparingly. The results in the paper show the high efficiency of the two methods as well as the abundant nonlinear phenomenon of the system, compared with the results obtained by the usual numerical solution of the Reynolds equation.