Order Basis Functions Research Articles

Exploring the stochastic patterns of hyperchaotic Lorenz systems with variable fractional order and radial basis function networks

Exploring the stochastic patterns of hyperchaotic Lorenz systems with variable fractional order and radial basis function networks

Grounded source transient electromagnetic 3D forward modeling with the spectral-element method and its application in hydraulic fracturing monitoring

A long wire with large current source transient electromagnetic (TEM) monitoring, with a large detection depth, low cost, safety, and environmental protection, has unique advantages in the testing and identification of unconventional reservoir fluid and the evaluation of stimulated reservoir volume. So, the TEM 3D forward modeling method has become a research hotspot. Although the finite-element method (FEM) is a type of numerical algorithm that has been widely applied in three-dimensional (3D) electromagnetic field forward modeling, the efficiency and accuracy of FEM require further improvement in order to meet the demand of fast 3D inversion. By increasing the order of the basis function and adjusting the principle of mesh discretization, the precision of the mixed-order spectral-element (SEM) result will be increased. The backward Euler scheme is an unconditionally stable technique which can ignore the impact of the scale of the time step. To achieve a better description of the nonlinear electromagnetic (EM) response of the grounded source TEM method and to optimize the efficiency and accuracy/precision of the 3D TEM forward modeling method significantly, we proposed the use of 3D TEM forward modeling based on the mixed-order SEM and the backward Euler scheme, which can obtain more accurate EM results with fewer degrees of freedom. To check its accuracy and efficiency, the 1D and 3D layered models are applied to compare the SEM results with the semi-analytical and FEM solutions. In addition, we analyzed the accuracy and efficiency of the SEM method for different types of order basis functions. Finally, we calculated the long-wire source TEM response for a practical 3D earth model of a shale gas reservoir for fracturing monitoring and tested the feasibility of the TEM method in a hydraulic fracturing monitoring area to further demonstrate the flexibility of the SEM method.

Estimating follow-up CTs from geometric deformations of catheter implants in interstitial breast brachytherapy: A feasibility study using electromagnetic tracking.

Electromagnetic tracking (EMT) systems have been shown to provide valuable information on the geometry of catheter implants in breast cancer patients undergoing interstitial brachytherapy (iBT). In the context of an extended patient-specific, pre-treatment verification, EMT can play a key role in determining the potential need and, if applicable, the appropriate time for treatment adaptation. To detect dosimetric shortcomings the relative position between catheters, and target volume and critical structures must be known. Since EMT cannot provide the anatomical context and standard imaging techniques such as cone-beam CT are not yet available in most brachytherapy suites, it is not possible to detect anatomic changes on a daily or fraction basis, so the need for adaptive planning cannot be identified. The aim of this feasibility study is to develop and evaluate a technique capable of estimating follow-up CTs at any time based on the initial treatment planning CT (PCT) and surrogate information about changes of the implant geometry from an EMT system. A deformation vector field is calculated from two different implant reconstructions acquired in treatment position through EMT, the first immediately after the PCT and the second at another time point during the course of treatment. The calculation is based on discrete displacement vectors of pairs of control and target points. These are extrapolated by means of different radial basis functions in order to cover the entire CT volume. The adequate parameters for the calculation of the deformation field were identified. By warping the PCT according to the deformation field, one obtains an estimated CT (ECT) that reflects the geometric changes. For the proof of concept, ECTs were computed for the time point of the clinical follow-up CT (FCT) that is embedded in the treatment workflow after the fourth fraction. ECT and clinical FCTs of 20 patients were compared to each other quantitatively in terms of absolute Hounsfield unit differences in the planning target volume (PTV) and in a convex hull (CH) enclosing the catheters. The median differences were 31.2 and 29.5HU for the CH and the PTV, respectively. The proposed ECT approach was able to approximate the "anatomy of the day" and therefore, in principle, allows a dosimetric appraisal of the treatment plan quality before each fraction. In this way, it can contribute to a more detailed patient-specific quality assurance in iBT of the breast and help to identify the timing for a potential treatment adaptation.

On an efficient octic order sub-parametric finite element method on curved domains

An efficient finite element method using the sub-parametric transformations of the curved boundaries for the 45-node octic order triangular element is presented in this paper. The one-sided curved and two-sided straight triangle in the universal coordinate system (x,y) is mapped onto an isosceles triangle (ξ,η)∈[0,1] and ξ+η≤1 in the (ξ,η)-coordinate system by using an isoparametric coordinate transformation. Using such a transformation, the curved edge of the triangular element is altered implicitly by a newly defined octic curve. Such an octic curve is obtained by using the topology of the curved edge. With an apparent condition of arc being a parabola, a new transformation is developed to find a unique curve for each edge that passes through the points on the original curved arc. Overall, the new sub-parametric transformations are developed, these include the transformation for the curved edge on the original boundary, in addition to the parameters from the curved triangle's interior. Several numerical experiments are presented to show the efficacy of the proposed method. The computational results obtained for a boundary value problem using octic order basis functions outperform the results obtained using the existing higher order basis functions such as quadratic (6-node), cubic (10-node), quartic (16-node), quintic (21-node), sextic (28-node), and septic (36-node) order triangular elements (TEs) with curves. The method implemented in this paper can be easily extendable to quasi-static and dynamic crack evolution problems within linear and nonlinear elasticity.

Stability of non-linear flow in heterogeneous porous media simulations using higher order and continuity basis functions

Using isogeometric analysis, we present a highly optimized FORTRAN code for simulations of three-dimensional non-linear flow problem in heterogeneous media. This includes deriving the weak formulations, discretizing with the forward Euler scheme, and implementing computation of the right-hand side. Some properties of exact solutions of heat transport and non-linear flow in heterogeneous media problems have been derived to validate the numerical solution. Stability analysis of the employed explicit time-stepping scheme has been performed. First, the properties of the discrete form of the heat transfer problem have been systematically investigated, leading to the conjecture concerning the sufficient condition of stability of the scheme, which numerical computations have subsequently confirmed. A similar analysis has been carried out for the non-linear flow in heterogeneous media problem. For this problem, the bound on the time step was conjectured to decrease exponentially, which has been verified experimentally. The simulation’s behavior close to the stability limit has been investigated and found to be considerably more complex than for the heat transport problem. Detailed experimental analysis of the time complexity of the particular parts of the simulator algorithm has been described. In addition, dependence on both mesh size and order of continuity of B-spline basis has been investigated.

Array Decomposition Method for Arbitrary-Element Regular Arrays Using Higher Order Basis Functions

The full-wave array decomposition method for regular antenna arrays with arbitrary elements using higher order hierarchical basis functions is investigated. We show that the use of higher order basis functions results in significantly reduced memory consumption and computation time for a <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\mathbf {10\!\times\!10}$</tex-math></inline-formula> element conical horn array with an aperture size of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$22\!\times\!22\lambda$</tex-math></inline-formula> , without the need for analytical nor numerical approximations. In addition, we demonstrate that by employing higher order basis functions, the far-field error is considerably lower than by using common first-order basis functions for the same total number of unknowns.

Stabilized Lagrange Interpolation Collocation Method: A meshfree method incorporating the advantages of finite element method

Since most approximation functions in meshfree methods are rational functions which do not possess the Kronecker delta property, how to achieve exact integration and accurately impose the essential boundary conditions are two typical difficulties for meshfree methods. In this paper, a new stabilized Lagrange interpolation collocation method (SLICM) is proposed in which the Lagrange interpolation (LI) is employed for the approximation in a meshfree method. This method can satisfy the high order integration constraints which can conserves the high order consistency conditions in the integration form. This property leads to the exact integration in the subdomains and optimal convergence for the proposed method. Meanwhile, performing the integration in subdomains can also reduce the condition number of discrete matrix, which improves the stability of the algorithm. Since the Lagrange interpolation approximation has Kronecker delta property, the essential boundary conditions can be simply and exactly imposed like the finite element method, which further improves the accuracy of this method. Convergence studies present that the same convergence rate can be attained for utilizing the odd and even order LI shape functions, while the convergence rate is reduced if the odd order basis function is employed in the reproducing kernel (RK) approximation. Numerical examples validate the high accuracy and convergence as well as good stability of the presented method, which can outperform the direct collocation method and the stabilized collocation method based on RK approximation.

ИССЛЕДОВАНИЕ ОБЛАСТИ ОПРЕДЕЛЕНИЯ ПАРАМЕТРОВ БАЗИСНОЙ ФУНКЦИИ ДВУХ АРГУМЕНТОВ ПРИ ПОСТРОЕНИИ Λ-ОРТОГОНАЛЬНОЙ БАЗИСНОЙ ФУНКЦИИ

The purpose of the research is to determine the area of setting the argument τ and its discretizationintervals when constructing an algorithm for adaptive nonlinearoptimal smoothing of multi-parameter data of trajectory measurements, which makes it possible tojointly implement the spatial and temporal redundancy of the data obtained. The research wascarried out by constructing a Λ-orthogonal basis function in order to obtain independent estimatesfor the coefficients of the smoothing polynomial. It is shown that it is advisable to solve the problemof determining the maximum likelihood estimate of the coefficient vector of the smoothingpolynomial by the method of successive approximations. When constructing a Λ-orthogonal basisfunction, the maximum likelihood estimate of the coefficient vector of the smoothing polynomial isachieved in 2-3 iterations. It follows from the research results presented in the paper that the accuracyindex Qт as a function of two arguments ( is the smallest value of the argument and Δτ isthe discretization interval of the argument τ) in a wide range of values of these arguments changesslightly, but increases sharply at . In this case, the values of these arguments should notexceed, respectively, the maximum and minimum possible numbers that can be written without lossof accuracy in the re grid of the computer used. With a uniform discretization step of the argument τ it is advisable to choose the argument in the middle part of the interval¸where andrespectively, are the minimum and maximum numbers that can be written into the bit grid of acomputer without loss of accuracy. In case of adverse conditions Approaching the edges of theinterval can lead to an increase in calculation errors in determining the secondary parameters ofthe position of the aircraft due to the fact that the main matrix of the system of equations becomesill-conditioned.

A Modified HODLR Solver Based on Higher Order Basis Functions for Solving Electromagnetic Scattering Problems

In this letter, we present a novel fast direct solver based on a modified hierarchically off-diagonal low-rank (M-HODLR) matrix format with higher order basis functions for analyzing electromagnetic scattering by electrically large objects. The M-HODLR matrix compresses the far-group submatrices by an extended admissibility condition, which enhances the efficiency and accuracy of the original HODLR. To further alleviate the burden of the fast direct solution of the conventional surface integral equation (SIE), the higher order hierarchical vector basis (HOHVB) functions defined over a curved surface are employed. To account for HOHVB, an efficient assembly scheme based on patch coupling is developed to accelerate the adaptive across approximation implementation in M-HODLR. The proposed method can significantly reduce the computational time and memory cost of a conventional SIE solver. Several numerical examples are presented to validate the accuracy and efficiency of the proposed method in solving electrically large objects. Owing to the characteristics of HOHVB, different patch sizes or basis functions with different orders can be chosen according to the local geometry of the object. Therefore, the proposed method is very appealing for large- and multiscale simulations.

Fundamental solutions for the Stokes equations: Numerical applications for 2D and 3D flows

We consider the application of the Method of Fundamental Solutions (MFS) to homogeneous force Stokes problems in 2 and 3 space dimensions. The choice of the main basis functions for the implementation of the MFS is justified by a new density result of linear combinations of Stokeslets in the L2-setting. This is convenient for Stokes flows with low degree of regularity which are found in many applications. In the case of mixed boundary conditions, Stresslets are added as basis functions in order to enforce the Neumann boundary condition. The accuracy of the method is investigated through a series of numerical tests, which include comparison between exact and numerical solutions and the application of the method to benchmark problems.

Detection of Lung Cancer Lesions Using 3D Convolutional Neural Networks and Segmentation for Accurate Detection

Lung cancer is a leading cause of death worldwide; it refers to the uncontrolled growth of abnormal cells in the lung. A computed tomography (CT) scan of the thorax is the most sensitive method for detecting cancerous lung nodules. A lung nodule is a round lesion which can be either non-cancerous or cancerous. In the CT, the lung cancer is observed as round white shadow nodules. In existing method, the candidate ROIs shape features are calculated, and some blood vessels are get rid of using rule-based according to shape features; secondly, the remainder candidates gray and texture features are calculated; finally, the shape, gray and texture features are taken as the inputs of the SVM (Support Vector Machine) classifier to classify the candidates. Experimental results show that the rule-based approach has no omission, but the misclassification probability is too large; Hence, in the proposed method the nodules were characterized by the computation of the texture features obtained from the gray level co-occurrence matrix (GLCM) in the wavelet domain and were classified using a SVM with radial basis function in order to classify CT images into two categories: with cancerous lung nodules and without lung nodules. The stages of the proposed methodology to design the CADx system are: 1) Extraction of the region of interest, 2) Wavelet transform, 3) Feature extraction, 4) Attribute and sub-band selection and 5) Classification. The same classification is implemented for the convolution neural networks. The final comparison is done between these two networks based on the accuracy.

Space-Time Isogeometric Analysis for linear and non-linear elastodynamics

In this study we investigate a stabilized space-time formulation for linear and non-linear elastodynamics. We use Isogeometric Analysis (IGA) in order to benefit from its numerical qualities. We focus on two points: the formulation of stabilized weak-forms in a linear and non-linear context and the interest of using continuous Galerkin schemes in space and time with higher order and higher continuity basis functions. We illustrate the numerical performance of these methods through typical impact or vibration problems commonly encountered in the field of the elastodynamics of solids.

Hybrid Multi-Channel EEG Filtering Method for Ocular and Muscular Artifact Removal Based on the 3D Spline Interpolation Technique

Abstract The present work develops a novel hybrid method for ocular and muscular artifact removal from electroencephalography (EEG) signals, EFICA-TQWT. It is a combination of efficient fast independent component analysis (EFICA) method with the tunable Q-factor wavelet transform (TQWT). The main contribution of this paper is to apply the 3D interpolation method in the filtering system. Three EEG datasets are used in this work, two healthy and one epileptic. The choice of subjects for each dataset is made with the help of an expert in physiology. The selection criterion adopted is the presence of muscular and ocular artifacts in the processed recordings. First, a noisy channel automatic classification is performed by the support vector machine (SVM) with radial basis function in order to delete the signal(s) corresponding to the noisiest channel(s) from each EEG recording. The results of the automatic classification by the SVM were compared with those found by the expert’s classification. An accuracy of 97.45%, a sensitivity of 86.66% and a 100% specificity are provided by the SVM classification. The hybrid method of artifact removal will be applied on the rest of the EEG channels of international 10/20 system for each subject. Then, a reconstruction of the eliminated channel signal(s) will be performed in order to obtain a well-filtered signal. The proposed filtering process is evaluated by calculating the mean squared error (MSE) and the signal to noise ratio (SNR). Both for the healthy and pathological EEG datasets, a comparative study of the proposed method (EFICA-TQWT) and other filtering techniques (Fast-ICA, DWT, TQWT and EFICA) is generated. The EFICA-TQWT method gave the best results with a minimum of MSE and a maximum of SNR, more particularly in the case of the application of the 3D interpolation method. Besides, in order to optimize the computing time of the proposed system, a parallel implementation of this filtering system is developed based on graphical processing units using compute unified device architecture.

Structural topology optimization using a level set method with finite difference updating scheme

In conventional level set methods for topology optimization, the evolution of level set function is solved by upwind scheme and needs to be reinitialized frequently throughout the optimization. The upwind scheme is complicated in numerical implementation and the reinitialization is time-consuming and slows down the optimization efficiency. This paper presents a level set method updated with finite difference scheme for structural topology optimization. The velocity field used to update the level set function is interpolated by piecewise basis function in order to incorporate the universal mathematical programming algorithm into the optimization. The diffusion is introduced into the Hamilton-Jacobi equation and the three-step splitting method is adopted to solve the equation. By applying the proposed method, the evolution of the level set can be updated by simple central difference scheme and the procedure of reinitialization is avoided without sacrificing the numerical stability. The classical mean compliance minimization problem with several numerical examples is used to test the validity of the proposed method.

Analysis of composite plates using isogeometric analysis: A discussion

Abstract Isogeometric analysis (IGA) concept is developed as the new numerical approach that reduces the gap between design and analysis by incorporating them both into a unified and single process. In this paper the application of this advanced numerical approach in analyzing composite plates based on different theories have been reviewed. Further, in the review development of different continuum theories based on IGA are also discussed. Firstly, incorporation of different equivalent single layer theories (ESL) with IGA have been discussed. Secondly, due to the consideration of linear displacement throughout the plates in ESL, discussion on IGA based layer wise theories (LWT) have been presented which includes non-linear displacements in each layer of composite plates. The main advantage behind incorporating IGA with these continuum theories is that the higher order and continuity basis function of IGA fulfils need of various continuum theories that results in an accurate and exact analysis of the composite plates. Also, the ease in increasing order of basis function in IGA helps to solve the composite plate problem very efficiently. Based on the advantages the calculated results of various IGA based studies were found to be in good agreement with the already existing numerical methods.

Mollified finite element approximants of arbitrary order and smoothness

The approximation properties of the finite element method can often be substantially improved by choosing smooth high-order basis functions. It is extremely difficult to devise such basis functions for partitions consisting of arbitrarily shaped polytopes. We propose the mollified basis functions of arbitrary order and smoothness for partitions consisting of convex polytopes. On each polytope an independent local polynomial approximant of arbitrary order is assumed. The basis functions are defined as the convolutions of the local approximants with a mollifier. The mollifier is chosen to be smooth, to have a compact support and a unit volume. The approximation properties of the obtained basis functions are governed by the local polynomial approximation order and mollifier smoothness. The convolution integrals are evaluated numerically first by computing the boolean intersection between the mollifier and the polytope and then applying the divergence theorem to reduce the dimension of the integrals. The support of a basis function is given as the Minkowski sum of the respective polytope and the mollifier. The breakpoints of the basis functions, i.e. locations with non-infinite smoothness, are not necessarily aligned with polytope boundaries. Furthermore, the basis functions are not boundary interpolating so that we apply boundary conditions with the non-symmetric Nitsche method as in immersed/embedded finite elements. The presented numerical examples confirm the optimal convergence of the proposed approximation scheme for Poisson and elasticity problems.

Adaptive 3D convolutional neural network-based reconstruction method for 3D coherent diffraction imaging

We present a novel adaptive machine-learning based approach for reconstructing three-dimensional (3D) crystals from coherent diffraction imaging. We represent the crystals using spherical harmonics (SH) and generate the corresponding synthetic diffraction patterns. We utilize 3D convolutional neural networks (CNNs) to learn a mapping between 3D diffraction volumes and the SH, which describe the boundary of the physical volumes from which they were generated. We use the 3D CNN-predicted SH coefficients as the initial guesses, which are then fine-tuned using adaptive model-independent feedback for improved accuracy. We also adaptively tune the locations, intensities, and decay rates of collections of radial basis functions in order to reproduce the non-uniform internal structure of 3D objects and demonstrate the method for a synthetic volume that has an internal void and a density ramp.

Maximally Orthogonalized Higher Order Basis Functions in Large-Domain Finite Element Modeling in Electromagnetics

Curl-conforming max-ortho basis functions (MOBFs) are coupled with higher order large-domain curved finite elements (FEs). The performance of the functions is compared with that of the classical and near-ortho basis functions. Through numerical experiments, it is shown that max-ortho FEs yield highly orthogonal mass matrices, for practically arbitrarily high orders of polynomial field approximations. This facilitates the usage of iterative solvers and it significantly increases their efficiency. Accurate and fast computation of MOBFs, of arbitrarily high orders, is enabled by the proposed two-term recurrent formula.

MAPL1: q-space reconstruction using -regularized mean apparent propagator.

To improve the quality of mean apparent propagator (MAP) reconstruction from a limited number of q-space samples. We implement an -regularised MAP (MAPL1) to consider higher order basis functions and to improve the fit without increasing the number of q-space samples. We compare MAPL1 with the least-squares optimization subject to non-negativity (MAP), and the Laplacian-regularized MAP (MAPL). We use simulations of crossing fibers and compute the normalized mean squared error (NMSE) and the Pearson's correlation coefficient to evaluate the reconstruction quality in q-space. We also compare coefficient-based diffusion indices in the simulations and in in vivo data. Results indicate that MAPL1 improves NMSE in 1 to 3% when compared to MAP or MAPL in a high undersampling regime. Additionally, MAPL1 produces more reproducible and accurate results for all sampling rates when there are enough basis functions to meet the sparsity criterion for the regularizer. These improved reconstructions also produce better coefficient-based diffusion indices for in vivo data. Adding an regularizer to MAP allows the use of more basis functions and a better fit without increasing the number of q-space samples. The impact of our research is that a complete diffusion spectrum can be reconstructed from an acquisition time very similar to a diffusion tensor imaging protocol.

Theoretical signature of a cavern created by an underground nuclear explosion in 2-D exploration seismic data

SUMMARYThe proliferation of nuclear tests is a problem that threatens the safety and health of everybody. In order to tackle this problem, the UN is promoting the Comprehensive Nuclear-Test-Ban Treaty (CTBT), which includes protocols for monitoring and On-Site Inspections (OSI). The purpose of OSI is to verify if a nuclear test has been carried out by identifying with a geophysical technique the presence of a cavern, hole or some device that indicates the violation of the treaty. In this context, it is desirable to be able to use exploration-seismology techniques to detect caverns created by an underground nuclear explosion. However, there is scarce information about the seismic signature of this type of cavern. We present the results of elastic wave propagation simulations, in the time domain, with a cavern created by an underground nuclear explosion. The wave equation is solved using the spectral element method with 4th order basis functions and quadrilateral elements. We show the results for models with cavities and caverns corresponding to explosions of 1, 20 and 100 kilotons, and obtain seismic traces in which we can observe the effect of the structures. We conclude that caverns created by nuclear explosions can be detected using seismic data and distinguished from caves because they behave like two concentric diffractor bodies, as opposed to caves-like diffractors.