scholarly journals The Spatial and Temporal Harmonic Balance Method for Obtaining Periodic Responses of a Nonlinear Partial Differential Equation With a Linear Complex Boundary Condition

2017 ◽  
Author(s):  
Xuefeng Wang ◽  
Weidong Zhu
Keyword(s):  
Boundary Condition ◽  
Harmonic Balance ◽  
Jacobian Matrix ◽  
Nonlinear Partial Differential Equation ◽  
System Matrix ◽  
Complex Boundary Condition ◽  
Complex Boundary ◽  
Toeplitz Form ◽  
Hill’S Method ◽  
Nonlinear String

The spatial and temporal harmonic balance (STHB) method is used to solve the periodic solution for a nonlinear partial differential equation (PDE) demonstrated by a nonlinear string equation with a linear complex boundary condition, and stablity analysis is conducted for the periodic solutions using Hill’s method. In order to avoid the integration procedure for discretizing the PDE to obtain the ordinary differential equations (ODEs), spatial and temporal harmonic balance procedures are conducted simultaneously, which can be efficiently achieved by the discrete sine transform and the fast Fourier transform. An additional coordinate associated with the generalized coordinates of the trial functions for the spatial discretization is introduced to make the solution satisfy all boundary conditions, and a relationship of the additional coordinate and the generalized coordinates is developed and used in the STHB method so that the test functions can be the same to the trial functions. Jacobian matrix of the harmonic balanced residual is obtained analytically, which can be used in Newton method for solving the periodic response. The STHB method and Jacobian matrix make the calculation of the periodic solution for the nonlinear string with a linear spring boundary condition efficient and easy to be implemented by computer programs. The relationship between the Jacobian matrix and the system matrix of the linearized ODEs are developed, so that one can directly obtain the Toeplitz form of the system matrix, and Hill’s method can be used to analyze the stability with the eigenvalues of the Toeplitz-form system matrix without the derivation of the ODEs. The frequency curve of the periodic solutions is obtained and their stability is indicated by the method in this work.

A Spatial and Temporal Harmonic Balance Method for Obtaining Periodic Solutions of a Nonlinear Partial Differential Equation With a Linear Complex Boundary Condition

2020 ◽  
Vol 142 (3) ◽  
Author(s):  
Xuefeng Wang ◽  
Weidong Zhu
Keyword(s):  
Boundary Condition ◽  
Periodic Solutions ◽  
Harmonic Balance ◽  
Jacobian Matrix ◽  
Additional Function ◽  
Complex Boundary Condition ◽  
Complex Boundary ◽  
Hill’S Method ◽  
Sine Functions ◽  
Nonlinear String

Abstract A spatial and temporal harmonic balance (STHB) method is demonstrated in this work by solving periodic solutions of a nonlinear string equation with a linear complex boundary condition, and stability analysis of the solutions is conducted by using the Hill’s method. In the STHB method, sine functions are used as basis functions in the space coordinate of the solutions, so that the spatial harmonic balance procedure can be implemented by the fast discrete sine transform. A trial function of a solution is formed by truncated sine functions and an additional function to satisfy the boundary conditions. In order to use sine functions as test functions, the method derives a relationship between the additional coordinate associated with the additional function and generalized coordinates associated with the sine functions. An analytical method to derive the Jacobian matrix of the harmonic balanced residual is also developed, and the matrix can be used in the Newton method to solve periodic solutions. The STHB procedures and analytical derivation of the Jacobian matrix make solutions of the nonlinear string equation with the linear spring boundary condition efficient and easy to be implemented by computer programs. The relationship between the Jacobian matrix and the system matrix of linearized ordinary differential equations (ODEs) that are associated with the governing partial differential equation is also developed, so that one can directly use the Hill’s method to analyze the stability of the periodic solutions without deriving the linearized ODEs. The frequency-response curve of the periodic solutions is obtained and their stability is examined.

Boundary Condition Updating of Bridge Structure Based on Mode Parameters

2010 ◽  
Vol 163-167 ◽  
pp. 3749-3756
Author(s):  
Zhou Shi ◽  
Ren Da Zhao ◽  
Shi Qiang Qin ◽  
Yu Feng Gao
Keyword(s):  
Boundary Condition ◽  
Constrained Optimization ◽  
Least Square ◽  
Inversion Method ◽  
Bridge Structure ◽  
Matrix Perturbation ◽  
Matrix Perturbation Theory ◽  
Complex Boundary Condition ◽  
Complex Boundary ◽  
Structure Boundary

According to the complex boundary condition of real bridge structure, appendix restraint parameters of structure boundary condition were detailed analyzed. The appendix horizontal and torsional restraint parameters were selected as research object. The least square constrained optimization inversion method was used to establish the question of reverse of bridge structure boundary appendix parameters based on tested and calculated mode parameters. The first order perturbation of eigen-value and eigen-vector based on matrix perturbation theory were substituted into the constrained optimization inversion equation, so the solving efficiency was improved greatly, and the iteration method was used to improve the precision. The corresponding program was made, and the example of a maglev railway girder shows that fairly well precise bridge structure boundary appendix parameters can be reserved with only former several modes parameters through four to seven times of iteration calculation.

An Efficient Galerkin Averaging-Incremental Harmonic Balance Method Based on the Fast Fourier Transform and Tensor Contraction

2020 ◽  
Author(s):  
R. Ju ◽  
W. Fan ◽  
W. D. Zhu
Keyword(s):  
Fourier Transform ◽  
Fast Fourier Transform ◽  
Harmonic Balance ◽  
High Efficiency ◽  
Jacobian Matrix ◽  
Sampling Rate ◽  
Harmonic Balance Method ◽  
Incremental Harmonic Balance ◽  
Tensor Contraction ◽  
Ihb Method

Abstract An efficient Galerkin averaging-incremental harmonic balance (EGA-IHB) method is developed based on the fast Fourier transform (FFT) and tensor contraction to increase efficiency and robustness of the IHB method when calculating periodic responses of complex nonlinear systems with non-polynomial nonlinearities. As a semi-analytical method, derivation of formulae and programming are significantly simplified in the EGA-IHB method. The residual vector and Jacobian matrix corresponding to nonlinear terms in the EGA-IHB method are expressed using truncated Fourier series. After calculating Fourier coefficient vectors using the FFT, tensor contraction is used to calculate the Jacobian matrix, which can significantly improve numerical efficiency. Since inaccurate results may be obtained from discrete Fourier transform-based methods when aliasing occurs, the minimal non-aliasing sampling rate is determined for the EGA-IHB method. Performances of the EGA-IHB method are analyzed using several benchmark examples; its accuracy, efficiency, convergence, and robustness are analyzed and compared with several widely used semi-analytical methods. The EGA-IHB method has high efficiency and good robustness for both polynomial and nonpolynomial nonlinearities, and it has considerable advantages over the other methods.

NONLINEAR APPROACH TO APPROXIMATE ACOUSTIC BOUNDARY ADMITTANCE IN CAVITIES

2007 ◽  
Vol 15 (01) ◽  
pp. 63-79 ◽  
Author(s):  
ROBERT ANDERSSOHN ◽  
STEFFEN MARBURG
Keyword(s):  
Boundary Condition ◽  
Sound Pressure ◽  
Pressure Field ◽  
Initial Boundary ◽  
System Matrix ◽  
Forward Solution ◽  
Piecewise Constant ◽  
Direct Inversion ◽  
Nonlinear Approach ◽  
Residual Norm

In this paper, an algorithm is derived to solve a problem of inverse acoustics. It considers the damped acoustic boundary value problem, i.e. the Helmholtz equation and admittance boundary condition, in order to approximate the boundary admittance of interior domains. The algorithm is implemented by using a finite element method and tested for two-dimensional cavities with arbitrary shapes. The admittance condition is reconstructed based on sound pressure measurements. The solution of the arising nonlinear system of equations is obtained by applying the Newton method following a presetting method for finding reasonable initial boundary admittance values. A residual norm accounts for the objective function. Its first- and second-order sensitivities are determined analytically by using a modal decomposition in order to avoid direct inversion of the system matrix. The experiment is simulated by taking sound pressure data of the forward solution as inputs for the inverse problem. Test examples show that very few measurement points are necessary to reproduce piecewise constant boundary admittance values very accurately. Then, the admittance boundary condition is applied to reproduce the sound pressure distribution in the cavity. Again, it becomes obvious that only a few measurement points are required to reconstruct the sound pressure field.

scholarly journals RBF-Based Meshless Method for Large Deflection of Elastic Thin Rectangular Plates with Boundary Conditions Involving Free Edges

2016 ◽  
Vol 2016 ◽  
pp. 1-10 ◽  
Author(s):  
Mohammed M. Hussein Al-Tholaia ◽  
Husain Jubran Al-Gahtani
Keyword(s):  
Boundary Condition ◽  
Basis Function ◽  
Meshless Method ◽  
Iterative Procedure ◽  
Large Deflection ◽  
Free Edge ◽  
Thin Plates ◽  
Rectangular Plates ◽  
Numerical Examples ◽  
Complex Boundary

An RBF-based meshless method is presented for the analysis of thin plates undergoing large deflection. The method is based on collocation with the multiquadric radial basis function (MQ-RBF). In the proposed method, the resulting coupled nonlinear equations are solved using an incremental-iterative procedure. The accuracy and efficiency of the method are verified through several numerical examples. The inclusion of the free edge boundary condition proves that this method is accurate and efficient in handling such complex boundary value problems.

On the Formulation of Nonreflecting Boundary Conditions for Turbomachinery Configurations: Part I — Theory and Implementation

2020 ◽  
Author(s):  
Christian Frey ◽  
Daniel Schlüß ◽  
Nina Wolfrum ◽  
Patrick Bechlars ◽  
Maximilian Beck
Keyword(s):  
Boundary Condition ◽  
Boundary Conditions ◽  
Frequency Domain ◽  
Harmonic Balance ◽  
Radial Component ◽  
Rotational Surface ◽  
Nonreflecting Boundary Conditions ◽  
Meridional Velocity ◽  
The Impact ◽  
Rotational Surfaces

Abstract With unsteady flow simulations of industrial turbomachinery configurations becoming more and more affordable there is a growing need for accurate inlet and outlet boundary conditions as numerical reflections alone can lead to incorrect trends in engine efficiency, noise and aeroelastic analysis parameters. This is the first of two papers on the formulation of unsteady boundary conditions which have been implemented for both time-domain and frequency-domain solvers. Giles’ original idea for steady solvers to formulate the boundary condition in terms of characteristics generalizes to frequency-domain solvers. The boundary condition drives the value of the incoming characteristics to ideal values that are computed using the modal decomposition of linearized 2D Euler flows. The present paper explains how to generalize 2D nonreflecting boundary conditions to real 3D annular domains by applying them in certain conical rotational surfaces. For a flow with zero radial component and an annular boundary that is perpendicular to the machine axis, these surfaces are the cylindrical streamsurfaces. For more general flows and geometries, however, there is no natural choice for the rotational surfaces. In this paper, two choices are discussed: the surfaces that are generated by the boundary normals and those that are defined by the circumferentially averaged meridional velocity. The impact of the boundary condition on the stability of the harmonic-balance solver is analyzed by studying the pseudo-time evolution of certain energy integrals. For a model problem which consists of a small disturbance of an inviscid flow, the increase or decrease of this energy integral is shown to be directly related to the normal characteristic variables along the boundary. This shows that the actual boundary condition should be formulated as a control problem for the normal characteristics. Moreover, the application of the harmonic balance solver to a simple duct configuration with prescribed disturbances demonstrates that using the characteristics based on the meridional velocity may prevent the solver from converging. In contrast, the 2D theory can be formulated in a different surface without impairing the robustness of the overall approach. These findings are illustrated by a simple test case. The impact of the choice of the rotational surface for the 2D theory is studied for various duct segments and a low-pressure turbine configuration in the second paper. There it is shown that applying the 2D theory to the meridional-velocity surfaces may be advantageous in that it leads to more accurate results.

A New Spatial and Temporal Harmonic Balance Method for Obtaining Periodic Steady-State Responses of a One-Dimensional Second-Order Continuous System

2016 ◽  
Vol 84 (1) ◽  
Author(s):  
X. F. Wang ◽  
W. D. Zhu
Keyword(s):  
Harmonic Balance ◽  
Nonlinear Term ◽  
Jacobian Matrix ◽  
Continuous System ◽  
Basis Functions ◽  
One Dimensional ◽  
Steady State Responses ◽  
Periodic Steady State ◽  
State Responses ◽  
Order Continuous

A new spatial and temporal harmonic balance (STHB) method is developed for obtaining periodic steady-state responses of a one-dimensional second-order continuous system. The spatial harmonic balance procedure with a series of sine and cosine basis functions can be efficiently conducted by the fast discrete sine and cosine transforms, respectively. The temporal harmonic balance procedure with basis functions of Fourier series can be efficiently conducted by the fast Fourier transform (FFT). In the STHB method, an associated set of ordinary differential equations (ODEs) of a governing partial differential equation (PDE), which is obtained by Galerkin method, does not need to be explicitly derived, and complicated calculation of a nonlinear term in the PDE can be avoided. The residual and the exact Jacobian matrix of an associated set of algebraic equations that are temporal harmonic balanced equations of the ODEs, which are used in Newton–Raphson method to iteratively search a final solution of the PDE, can be directly obtained by STHB procedures for the PDE even if the nonlinear term is included. The relationship of Jacobian matrix and Toeplitz form of the system matrix of the ODEs provides an efficient and convenient way to stability analysis for the STHB method; bifurcations can also be indicated. A complex boundary condition of a string with a spring at the boundary can be handled by the STHB method in combination with the spectral Tau method.

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