M. S. Kalra

Error estimates for tomographic inversion

The technique of computerized tomography is studied extensively by engineers as well as mathematicians to improve the quality of reconstructed images. Certain error estimates are available for the reconstruction errors occurring in various tomographic algorithms, in particular the convolution backprojection (CBP) method. The authors present an attempt toward developing some error estimates for predicting the inherent error due to the band-limiting assumption incorporated in the CBP methodology. The norms used are local in character compared to the general norms used in previous studies.

Error analysis of tomographic filters II: results

Abstract The convolution backprojection algorithm (CBP) has been studied from the point-of-view of the already published error formula which relates the error in reconstructions, under certain ideal conditions, to the Fourier-space derivatives of the filter functions employed in CBP. The results, for simulated images representing a damaged nuclear reactor fuel assembly and a cross-section of human brain, indicate that the error pattern obtained is in concurrence with the theory. Experimental results for a pixel size of 20 μm are also included.

Time integration in the dual reciprocity boundary element analysis of transient diffusion

Abstract This paper presents a comprehensive study of the time integration in the dual reciprocity boundary element analysis of transient diffusion. Detailed numerical experiments are performed using four representative test problems to assess the stability and presence of numerical oscillations, convergence rate, and the time response of various time integration algorithms, viz. one and two step least squares methods, cubic Hermitian schemes, and one step and multistep θ-methods. A discussion of computational aspects such as the effect of flux averaging for Dirichlet problems, starting procedure for multistep methods and the computational efficiency is also included. The results indicate that for Dirichlet problems, a one step backward difference method should be preferred for short term response, whereas higher order schemes should be used for long term response. For problems free from Dirichlet boundary conditions, all the higher order schemes yield accurate results over the entire time domain. For all the problems, a one step least squares algorithm appears to be the most accurate and efficient for medium to long term response. Further, an alternative time integration approach, which involves the partitioning of the boundary element system into differential and algebraic components, is proposed. Numerical results indicate that the partitioned approach effectively damps out spurious numerical oscillations and results in more accurate solutions than the regular approach in which the differential algebraic system is solved in the usual way without partitioning.

Least squares finite element formulation in the time domain for the dual reciprocity boundary element method in heat conduction

Abstract This paper presents a least squares finite element scheme applied to the time domain in the dual reciprocity BEM as an alternative to two-point finite difference or weighted residual schemes which have been used so far in the literature to integrate the system of ordinary differential equations in time arising from the spatial boundary element discretization. The proposed scheme obtains the desired recurrence relations via a least squares formulation in the context of one linear time element representing the entire time domain. Global boundary errors are used to obtain a measure of solution accuracy and convergence behavior of the proposed scheme through detailed and systematic numerical experiments. Results are presented for four representative problems and compared with those obtained using three weighted residual schemes viz. the fully implicit, the Galerkin and the Crank-Nicolson schemes. The least squares scheme obtains, in general, the most accurate results at about the same computational cost as the weighted residual schemes and exhibits superior convergence behavior. With linear time elements, the proposed scheme shows nearly quadratic rate of convergence for all the problems considered; it indicates exponential convergence for the problems involving Dirichlet boundary conditions. Further, this least squares scheme gives very accurate large time solutions.

Mart Algorithms for Circular and Helical Cone-Beam Tomography

The present work is concerned with the evaluation of the performance and the efficient implementation of multiplicative algebraic reconstruction technique (MART) to reconstruct three-dimensional (3D) objects for two different source/detector trajectories. Three types of MART algorithms are tested on a numerical phantom (Defrise), and they are implemented on a 3D X-ray system of Vikram Sarabhai Space Centre (VSSC). Circular and helical cone-beam trajectories are used. The results are compared with convolution backprojection (CBP) algorithm for each trajectory. It is found that iterative algorithms perform better than their counterpart, the transform-based CBP algorithm, whenever tomography systems are ill-conditioned due to limited views and/or noisy projection data.

Least-squares finite element schemes in the time domain

Least-squares finite element procedure is used to generate recurrence relations for numerical solution of system of ordinary differential equations of the first order. One-step least-squares method due to O.C. Zienkiewicz and R.W. Lewis [Earthquake Engrg. Struct. Dyn. 1 (1973) 407–408] is reviewed. An analysis of stability and other numerical properties of this method is presented, and it is found to be A-stable and second-order accurate. Using quadratic time element with Lagrange interpolation functions, a two-step least-squares method is derived. Analysis of local discretization error, stability and other properties of the two-step method is presented. It is found that the two-step least-squares algorithm is third-order accurate and A(α)-stable (α≈85∘). Comparison of numerical results obtained with the least-squares schemes with those obtained with other well known algorithms shows that the two-step least-squares scheme and three-step backward-difference scheme exhibit almost the same accuracy, whereas the one-step least-squares scheme is more accurate than one-step θ-methods and two-step backward-difference scheme. Further, the least-squares schemes exhibit superior accuracy at large time values for the problems tending towards a steady state.

Characteristic Signature of Specimen Using an Approximate Formula for 3D Circular Cone-Beam Tomography

The approximate nature of the Feldkamp–Davis–Kress (FDK) algorithm for circular cone-beam tomography motivates the error estimation in the reconstruction of three-dimensional (3D) objects. This algorithm is based on 3D cone-beam backprojection and 1D band-limiting filtering. The use of different window functions along with band-limiting filter in the convolution step of this algorithm lead to different reconstructions for the same projection data set, and a theoretical error gets incorporated in the reconstruction process. The present study is an attempt to understand this error in the reconstructed images from FDK algorithm using 3D version of “First Kanpur theorem” (KT-1). This theorem was proposed initially for error estimates in 2D reconstructions, and here it is extended to cone-beam volume reconstructions. The 3D version of KT-1 for circular cone-beam reconstructions is validated using industrial and medical numerical phantoms. It is then implemented on variety of real-life specimens. A characteristic signature is evolved as an application of the Kanpur Theorem (KT-1), which is a unique representation for a volumetric object. The preliminary obtained results are consistent and quite encouraging.

Three and four step least squares finite element schemes in the time domain

The paper presents a formulation and analysis of three and four step least squares algorithms for first order IVPs. The three step algorithm is derived using cubic Lagrangian interpolation, and is found to be third order accurate but only conditionally stable. Fourth order Lagrangian interpolation is used to obtain a four step least squares scheme which is A 0 -stable but inconsistent.

Application of MFS-MPS to the current-hole simulation in a Tokamak

A meshless method based on fundamental and particular solutions (MFS-MPS) has been implemented for the current-hole simulation in cylindrical tokamaks. We first benchmark the method by solving the Grad-Shafranov (GS) equation for zero as well as nonzero inverse aspect ratios and noncircular cross-sections. Thereafter, the method is implemented to solve the time-dependent reduced resistive MHD equations for the current-hole simulation. The initial current density profile with a negative current density near the center is chosen and the corresponding equilibrium magnetic flux profile is first computed using the GS equation. These profiles are then used to solve the coupled nonlinear system of MHD equations using MFS-MPS and a semi-implicit time differencing scheme. After an initial linear phase extending up to a few thousand Alfven times, the nonlinear oscillations are observed which continue for several Alfven times. This oscillating behavior is in agreement with the previous simulations using other numerical methods. It is also found that these oscillations are damped out at higher resistivity.

Steam-generator simulation with non-equilibrium two-phase flow models

Abstract The steam-generator model initially developed was a five-equation simulation of the transient behaviour. The present study extends the earlier work by using six field equations to simulate the two-phase flow conditions in the secondary side of the steam generator. Two versions of the six-equation model have been tested. No numerical instabilities were observed. The transient predictions with the two versions were almost identical for the PWR data used in this study.