Conjugate Gradient Fast Fourier Transform Research Articles

Assembly accuracy analysis and phase optimization of aero-engine multistage rotors considering surface morphology and non-uniform contact deformation

The assembly accuracy of multistage rotors is crucial to guarantee the quality and performance of aero engines. However, traditional assembly accuracy analysis methods usually oversimplify geometric deviations as the degrees of freedom (DOF) variations relative to the ideal position, ignoring real surface morphology and the resulting non-uniform contact deformation. Therefore, this paper proposes an assembly accuracy analysis method that integrates the effects of surface morphology and non-uniform contact deformation on error propagation process, and further adopts a phase optimization method to improve the assembly accuracy of multistage rotors. Specifically, the surface morphology of rotor end faces is generated using the random midpoint displacement method. The Conjugate Gradient-Fast Fourier Transform (CG-FFT) algorithm is employed to calculate the non-uniform contact deformation. According to the simulation results, a fitting deformation surface (FDS) is constructed to determine the assembly position and orientation. On this basis, a novel error propagation model is established and the mounting phase of each stage rotor is optimized by a genetic algorithm (GA). A case study of a specific rotor system is carried out to verify the proposed error propagation model and give the optimized assembly combination with the objective of minimum coaxiality. This research effectively improves the prediction and actual assembly accuracy of aero-engine multistage rotors and provide a theoretical basis for the precision assembly of multistage rotors in aero-engine applications.

Rapid Subentire-Domain Basis Functions Method Based on Adaptive Artificial Neural Networks

The subentire-domain (SED) basis functions method has shown its efficiency in solving the electromagnetic problem of large-scale finite periodic structures (LFPSs). However, calculating the expansion coefficients of SED basis functions is very time-consuming due to the consideration of the mutual coupling between all the elements in LFPSs, even after accelerated by conjugate-gradient fast Fourier transform (CG-FFT) and/or fast multipole method (FMM). In this article, based on the physics locations of observation cells, the adaptive artificial neural networks (AANNs) have been employed to rapidly predict the expansion coefficients of SED basis functions on interior cells, edge cells, and corner cells. By involving the AANNs, the mutual coupling between all the elements can be accounted into the neural networks without the construction of the mutual coupling matrix, and the expansion coefficients of SED basis functions can be obtained rapidly. Numerical experiments prove the accuracy and efficiency of the AANN-assisted SED basis functions method.

On the computational complexity of the conjugate-gradient method for solving inverse scattering problems

This paper presents an efficient implementation of the inversion algorithm based on the conjugate-gradient method (CGM) for solving inverse scattering problems. The original CGM provides good image reconstruction and robustness to noise-corrupted data. This method requires the solution of the forward scattering problem and the Fréchet derivative operator of the cost function at each iteration step. However, these procedures can make the computational cost prohibitive, even for moderately sized problems. To avoid the computational burden, a two-step conjugate gradient fast Fourier transform (CG-FFT) procedure is proposed. Such an approach reduces the computational cost and storage requirements of the CGM implementation. The efficient CGM is found to share a computational complexity similar to the distorted-Born iterative method (DBIM). Thus the convergence speed and accuracy of the CGM is compared with the DBIM. Numerical tests using both synthetic and experimental data show effectiveness for solving 2D inverse scattering problems.

Considerations of form defects and surface deformations for tolerance analysis of cylindrical components

The modeling and simulation of cylindrical surfaces with consideration of form defects have led to considerable research outcomes in the field of Computer-Aided Tolerancing (CAT). However, further consideration of surface deformations caused by external forces still remains a challenge. To address this issue, this paper properly considers the form defects and surface deformations for tolerance analysis of cylindrical components. First, form defects are considered by modeling skin model shapes of cylindrical surfaces. Afterwards, the tight fit and loose fit of a pair of cylindrical surfaces are identified, and the simulation methods of their positioning are presented. Specifically, for tight fit situation, a k-d tree based Iterative Closest Point (ICP) algorithm is used, and for loose fit situation, the constrained registration approach is adopted. Moreover, a Conjugate Gradient-Fast Fourier Transform (CG-FFT) method is presented for the consideration of surface deformations. In addition, simulations of given examples are conducted, which show the considerable effects of form defects and surface deformations. The simulations may also help determine the best performance of the to-be-assembled cylindrical components.

Polytope-based tolerance analysis with consideration of form defects and surface deformations

ABSTRACT Tolerance analysis aims to evaluate the functional requirement of mechanical assemblies in the design stage. As one of the typical tolerance analysis methods, the polytope model has shown great potential in Computer-Aided Tolerancing (CAT) field based on a series of constraints. However, existing research outcomes on this topic rarely consider form defects and the combined effects of form defects and surface deformations. This limitation may cause inaccuracy of tolerance analysis results. Hence, this paper integrates form defects and surface deformations in polytope-based tolerance analysis. Specifically, form defects are considered based on the concept of Skin Model Shapes and surface deformations are calculated based on a Conjugate Gradient-Fast Fourier Transform (CG-FFT) method. Then, the modeling method of the new polytope model is elaborated based on surface types and different types of constraints. Moreover, a case study is conducted to explain the procedure of polytope-based tolerance analysis. Through operations of the polytopes, tolerance analysis results are obtained, and the comparison with conventional polytope analysis results shows the considerable effects of form defects and surface deformations. It is expected that the proposed method can have a more realistic and accurate analysis result because of its consideration of real surface contact status.

Integration of surface deformations into polytope-based tolerance analysis: application to an over-constrained mechanism

Abstract Polytope-based tolerance analysis has shown a great potential in the Computer Aided Tolerancing (CAT) field. Previous works have considered form defects into this model, which overcomes the drawback of considering toleranced features as surfaces of perfect form. To enrich the capability and applicability of conventional polytope model, this paper further integrates the combined effects of form defects and surface deformations into polytope-based tolerance analysis. Specifically, form defects are considered based on the concept of Skin Model Shapes and surface deformations are calculated based on a Conjugate Gradient Fast Fourier Transform (CG-FFT) method. Thereafter, the polytope-based tolerance analysis is applied to an over-constrained mechanism. With the consideration of non-ideal surface contact status, this new polytope model provides more realistic and accurate analysis results.

Effects of contact behaviors on tolerance analysis of mechanism based on skin model shapes and a boundary element method

Abstract Tolerance analysis plays an important role in predicting the quality of mechanical products in early design stage. However, previous researches on tolerance analysis have mostly ignored the effects of contact behaviors between mating surfaces. This may lead to unrealistic results, especially when analyzing products with kinematic pairs such as rotating mechanisms. In order to obtain more accurate and realistic analysis results, this paper proposes a tolerance analysis approach that considers the effects of contact behaviors between mating surfaces. Geometric features with machining and assembly errors are modeled based on the concept of Skin Model Shapes. Contact behaviors between mating surfaces are calculated by the Conjugate Gradient-Fast Fourier Transform (CG-FFT) method, a classic Boundary Element Method (BEM) for solving contact problems. On these bases, a cam mechanism is adopted as an example to illustrate the effects of contact behaviors on transmission performance. Through the tolerance analysis, not only the transmission error of cam mechanism is analyzed, but also the contact status and pressure distribution between kinematic pairs can be obtained, which provide references for other aspects such as reliability analysis.

The Effect of Anisotropy on the Percolation Threshold of Sealing Surfaces

The percolation threshold strongly affects sealing performance. This paper investigates the relationship between the percolation threshold and the rough surface anisotropy, which is represented by the Peklenik number, γ. A series of anisotropic rough surfaces were generated and the conjugate gradient-fast Fourier transform (CG-FFT) method was used to determine the percolation threshold. The percolation threshold was found to be A/A0≈0.484±0.009 (averaged over 45 surfaces) was established for an isotropic rough surface (γ=1). Furthermore, it was also found that the percolation threshold decreased from A/A0≈0.528±0.011 to A/A0≈0.431±0.008 as 1/γ increased from 0.6 to 2. Our results differ from the theoretical result of Persson et al., where A/A0=γ/(1+γ). Comparing our calculated results with the theoretical results established the presence of an intersection value of 1/γ that was related to the effect of elastic deformation on the percolation threshold. When 1/γ was smaller than the intersection value, our calculated results were lower than the theoretical ones; and when 1/γ was greater than the intersection value, our calculated results were higher than the theoretical ones.

Integrating form errors and local surface deformations into tolerance analysis based on skin model shapes and a boundary element method

Tolerance analysis aims to ensure the performance of mechanical products. However, traditional analysis methods largely ignore the effects of form errors and local surface deformations, which are inevitably introduced during manufacturing and assembly phases. Therefore, this paper proposes a novel tolerance analysis framework integrating form errors and local surface deformations based on skin model shapes and a Boundary Element Method (BEM) for rectangular surfaces. First, it introduces a method for modeling skin model shapes considering form errors for rectangular surfaces. Next, it presents the calculation of local surface deformations based on the Conjugate Gradient-Fast Fourier Transform (CG-FFT) method, a classic BEM method for solving contact problems. Together, these can be used to obtain the relative positioning of mating surfaces considering the effects of form errors and local surface deformations. Case studies revealed the considerable effects of form errors and local surface deformations on assembly deviations, and demonstrated the efficiency of the proposed tolerance analysis process. Because the proposed tolerance analysis framework couples geometric deviations introduced during the manufacturing phase with mechanical deformations introduced during the assembly phase, it is expected to help control the geometric deviations of mechanical assemblies more accurately and also ensure product performance.

Tolerance analysis of annular surfaces considering form errors and local surface deformations

This paper proposes a tolerance analysis method of annular surfaces considering form errors and local surface deformations. Specifically, form errors are properly considered based on skin model shapes. Local surface deformations are calculated based on a Conjugate Gradient-Fast Fourier Transform (CG-FFT) method, which is a classic Boundary Element Method (BEM). Based thereon, tolerance analysis of annular surfaces is conducted based on Monte Carlo simulation, and efficiency of the process is also demonstrated.

Fast Fourier Transform Based Iterative Method for Electromagnetic Scattering From 1D Flat Surfaces

An efficient iterative method is proposed for computing the electromagnetic fields scattered from a one dimensional (1D) flat surface. The new iterative method is based on a similar implementation to the Conjugate Gradient Fast Fourier Transform (CG-FFT), where acceleration of the matrix-vector multiplications is achieved using fast Fourier transforms (FFT). However, the iterative method proposed is not based on Krylov subspace expansions and is shown to converge faster than GMRES-FFT and CGNE-FFT while maintaining the computational complexity and memory usage of those methods. Analysis is presented deriving an explicit convergence criterion.

A contrast-source integral-equation approach for three-dimensional modeling of elastic wave problems

In this paper, a recently proposed formulation of an integral equation for solving three-dimensional elastic wave scattering problems is numerically implemented. The approach is formulated in terms of the stress tensor and particle velocity vector, where the symmetric tensors of rank two are decomposed into their omnidirectional and deviatoric constituents. Subsequently, this integral equation is used to obtain a contrast-source type integral equation. For solving these integral equations we employ a Conjugate Gradient Fast Fourier Transform (CG-FFT) scheme, which is based on quadrature formulas that provide (second-order) accurate approximations while retaining the convolution nature of the relevant integrals that make them amenable to efficient evaluation via Fast Fourier Transforms. As linear solvers we employ the Conjugate Gradient for Normal Residual (CGNR) scheme, which is always monotonically convergent, but has a slow convergent rate, and the Bi-Conjugate Gradient Stabilized (BiCGSTAB) scheme, which is more efficient, but it is less stable. The convergence rates of iterative schemes are further improved through the use of a simple diagonal preconditioner. Numerical results are presented to demonstrate the accuracy and efficiency of the proposed approaches.

A CG-FFT approach to the solution of a stress-velocity formulation of three-dimensional elastic scattering problems

In this paper we introduce a Conjugate Gradient Fast Fourier Transform (CG-FFT) scheme for the numerical solution of the integral equation formulating three-dimensional elastic scattering problems. The formulation is in terms of the stress tensor and particle velocities as the unknown field variables. In contrast with the formulation based on particle displacements, this approach leads to integral representations that do not involve derivatives of the unknown fields, thus resulting in simplified and more stable numerics. The numerical procedure is based on suitable quadrature formulas that provide (second order) accurate approximations while retaining the convolution nature of the relevant integrals that make them amenable to efficient evaluation via FFTs. The scheme is further improved through the introduction of (approximation-based) pre-conditioners that are shown to accelerate the convergence of the CG iterations. Numerical results are presented that demonstrate the accuracy and efficiency of the proposed methodology.

Analysis of Electromagnetic Scattering from Plasma Antenna Using CG-FFT Method

A gaseous plasma column as an efficient radiator of electromagnetic waves is well known. This paper presents the scattering analysis of a plasma antenna using the conjugate gradient fast Fourier transform (CG-FFT) method. Radar cross sections (RCS) of designed plasma antenna for different plasma parameters are computed mainly. Numerical results show that the plasma antenna with appropriate parameters can offer lower RCS than the metal one. It is evident from the observed scattering characteristics that the plasma antenna can be regarded as a rod of imperfect conductor with losses.

Application of the Spatial–Spectral CG-FFT Method for the Solution of Electromagnetic Scattering by Buried Flat Metallic Objects

The conjugate gradient fast Fourier transform (CG-FFT) method to analyze electromagnetic scattering by buried flat metallic objects of arbitrary shape and large size is presented. Due to the use of FFT in the spatial domain and spectral domain to handle a spatial-domain convolution, the electric field integral equation and scattered fields can be rapidly solved without evaluations of Sommerfeld integrals. The accuracy of this algorithm is better than that of the conventional CG-FFT method, and the CPU time required for this algorithm is reduced to a minimum, while memory cost is order of N (the total cell number) and computational complexity is of order N log N in each iteration

Analysis of Large-Scale Periodic Array Antennas by CG-FFT Combined with Equivalent Sub-Array Preconditioner

SUMMARY This paper presents method that offers the fast and accurate analysis of large-scale periodic array antennas by conjugate-gradient fast Fourier transform (CG-FFT) combined with an equivalent sub-array preconditioner. Method of moments (MoM) is used to discretize the electric field integral equation (EFIE) and form the impedance matrix equation. By properly dividing a large array into equivalent sub-blocks level by level, the impedance matrix becomes a structure of Three-level Block Toeplitz Matrices. The Three-level Block Toeplitz Matrices are further transformed to Circulant Matrix, whose multiplication with a vector can be rapidly implemented by one-dimension (1-D) fast Fourier transform (FFT). Thus, the conjugate-gradient fast Fourier transform (CG-FFT) is successfully applied to the analysis of a large-scale periodic dipole array by speeding up the matrix-vector multiplication in the iterative solver. Furthermore, an equivalent sub-array preconditioner is proposed to combine with the CG-FFT analysis to reduce iterative steps and the whole CPU-time of the iteration. Some numerical results are given to illustrate the high efficiency and accuracy of the present method.

Fast analysis of microwave-integrated circuits using the loose GMRES-FFT method: Research Articles

In this article, the microstrip circuit is analyzed in terms of the mixed-potential integral equation (MPIE) by means of the rooftop-function expansion and the blaze-function testing technique, and the integral equation is solved using the loose generalized minimal residual fast Fourier transform (LGMRES-FFT) method. Our numerical calculations show that LGMRES-FFT can converge faster than the conjugate gradient-fast Fourier transform (CG-FFT) method. Some typical microstrip discontinuities are analyzed and the good results obtained demonstrate the validity of the proposed algorithm. © 2005 Wiley Periodicals, Inc. Int J RF and Microwave CAE, 2005.

Fast algorithms for large-scale periodic structures using subentire domain basis functions

Two efficient algorithms are proposed to analyze a large-scale periodic structure with finite size using the subentire-domain (SED) basis functions and the conjugate-gradient fast Fourier transform (CG-FFT). The SED basis function is defined on the support of each single element of the periodic structure. In a simplified SED (SSED)-CG-FFT algorithm, all elements of the periodic structure share the same SED basis function. As a consequence, SSED-CG-FFT can be performed in the whole periodic structure. However, SSED-CG-FFT becomes less accurate if the gap between two unit elements is very small, where the single SED basis function cannot capture the strong mutual coupling. In order to consider the mutual coupling, an accurate SED (ASED)-CG-FFT algorithm is proposed. In this algorithm, nine types of SED basis functions are employed to distinguish interior cells, edge cells, and corner cells. As a consequence, ASED-CG-FFT can be performed in all interior cells of the periodic structure. Comparing with the conventional method of moments with subdomain basis functions, the proposed algorithms are more efficient in both the computational complexity and the memory requirement. Numerical results are given to test the validity and efficiency of the proposed methods.

Fast analysis of microwave-integrated circuits using the loose GMRES-FFT method

In this article, the microstrip circuit is analyzed in terms of the mixed-potential integral equation (MPIE) by means of the rooftop-function expansion and the blaze-function testing technique, and the integral equation is solved using the loose generalized minimal residual fast Fourier transform (LGMRES-FFT) method. Our numerical calculations show that LGMRES-FFT can converge faster than the conjugate gradient-fast Fourier transform (CG-FFT) method. Some typical microstrip discontinuities are analyzed and the good results obtained demonstrate the validity of the proposed algorithm. © 2005 Wiley Periodicals, Inc. Int J RF and Microwave CAE, 2005.

Fast analysis of microwave integrated circuits by use of the inner‐outer flexible GMRES‐FFT method

AbstractIn this paper, the microstrip circuit is analyzed in terms of the mixed potential integral equation (MPIE) by using the rooftop‐function expansion and the blaze‐function testing technique. The integral equation is solved by the inner‐outer flexible generalized minimal residual fast Fourier transform (FGMRES‐FFT) method. Our numerical calculations show that the FGMRES‐FFT method can converge as high as five times faster than the generalized minimal residual‐fast Fourier transform (GMRES‐FFT) method and 16 times faster than the conjugate gradient‐fast Fourier transform (CG‐FFT) method. Some typical microstrip discontinuities are analyzed and the good results obtained demonstrate the validity of the proposed algorithm. © 2004 Wiley Periodicals, Inc. Microwave Opt Technol Lett 43: 409–413, 2004; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/mop.20485