Finite-Element Neural Network-Based Solving 3-D Differential Equations in MFL

The solution of a differential equation contains the forward model and the inverse problem. The finite element method (FEM) and the iterative approach based on FEM are extensively used to solve varied differential equations. Although FEM could obtain an accurate solution, the shortcoming of the approach is the high computational costs. This paper proposes an improved finite-element neural network (FENN) embedding a FEM in a neural network structure for solving the forward model while a conjugate gradient (CG) method is employed as the learning algorithm. Taking the 3-D magnetic field analysis in magnetic flux leakage (MFL) testing as an example, the comparison between CG algorithm and the gradient descent (GD) algorithm is presented. The vector plot of magnetic field intensity is obtained, and the vertical components of magnetic flux density are respectively analyzed. The iterative approach based on FENN and parallel radial wavelet basis function neural network is also adopted to solve the inverse problem. This approach iteratively adjusts weights of the inverse network to minimize the error between the measured and predicted values of MFL signals. The forward and inverse results indicate that FENN and the iterative approach are feasible methods with rapidness, accuracy and stability associated with 3-D different equations in MFL testing.

[1]  Gui Yun Tian,et al.  Numerical simulation on magnetic flux leakage evaluation at high speed , 2006 .

[2]  L. Udpa,et al.  Adaptive Wavelets for Characterizing Magnetic Flux Leakage Signals from Pipeline inspection , 2006, INTERMAG 2006 - IEEE International Magnetics Conference.

[3]  Yukio Kosugi,et al.  Neural network representation of finite element method , 1994, Neural Networks.

[4]  Shiyu Sun,et al.  Application of 3-D FEM in the simulation analysis for MFL signals , 2009 .

[5]  Kenzo Miya,et al.  Reconstruction of crack shapes from the MFLT signals by using a rapid forward solver and an optimization approach , 2002 .

[6]  J. Reilly,et al.  Sizing of 3-D Arbitrary Defects Using Magnetic Flux Leakage Measurements , 2010, IEEE Transactions on Magnetics.

[7]  D. Lowther,et al.  The Application of Topological Gradients to Defect Identification in Magnetic Flux Leakage-Type NDT , 2010, IEEE Transactions on Magnetics.

[8]  G. Capizzi,et al.  A neural network approach for the differentiation of numerical solutions of 3-D electromagnetic problems , 2004, IEEE Transactions on Magnetics.

[9]  Satish S. Udpa,et al.  Finite-element neural networks for solving differential equations , 2005, IEEE Transactions on Neural Networks.

[10]  Lynann Clapham,et al.  A model for magnetic flux leakage signal predictions , 2003 .

[11]  Carretera de Valencia,et al.  The finite element method in electromagnetics , 2000 .

[12]  Satish S. Udpa,et al.  Characterization of gas pipeline inspection signals using wavelet basis function neural networks , 2000 .

[13]  Salvatore Coco,et al.  A multilayer perceptron neural model for the differentiation of Laplacian 3-D finite-element solutions , 2003 .

[14]  Chao Xu,et al.  Reconstruction of 3D defect profiles from the MFLT signals by using a radial wavelet basis function neural network iterative model , 2012 .

[15]  Xinjun Wu,et al.  Local area magnetization and inspection method for aerial pipelines , 2005 .

[16]  Lalita Udpa,et al.  Neural network-based inversion algorithms in magnetic flux leakage nondestructive evaluation , 2003 .

[17]  A C Bruno,et al.  Experimental verification of a finite element model used in a magnetic flux leakage inverse problem , 2005 .

[18]  R. K. Stanley,et al.  Simulation and Analysis of 3-D Magnetic Flux Leakage , 2009, IEEE Transactions on Magnetics.

[19]  Kazufumi Kaneda,et al.  Direct solution method for finite element analysis using Hopfield neural network , 1995 .

[20]  R. K. Stanley,et al.  Dipole Modeling of Magnetic Flux Leakage , 2009, IEEE Transactions on Magnetics.

[21]  R. C. Ireland,et al.  Finite element modelling of a circumferential magnetiser , 2006 .

[22]  Lalita Udpa,et al.  Electromagnetic NDE signal inversion by function-approximation neural networks , 2002 .