Hybrid Method of Engineering Analysis: Combining Meshfree Method With Distance Fields and Collocation Technique

2009 ◽  
Author(s):  
Igor Tsukanov ◽  
Sudhir R. Posireddy
Keyword(s):  
Boundary Conditions ◽  
Numerical Technique ◽  
Geometric Model ◽  
Meshfree Method ◽  
Basis Functions ◽  
Engineering Analysis ◽  
Alternative Analysis ◽  
Distance Fields ◽  
Analysis Methods ◽  
Collocation Technique

Most of the modern engineering analysis methods (Finite Element, Finite Difference, Finite Volume, etc.) rely on space discretizations of the underlying geometric model. Such spatial meshes have to conform to the geometric model in order to approximate boundary conditions, construct basis functions with good local properties as well as perform numerical integration and visualization of the modeling results. Despite recent advances in automatic mesh generation, spatial meshing still remains difficult problem which often requires geometry simplification and feature removal. Conforming spatial mesh also restricts motions and variations of the geometry and breaks design-analysis cycle. In order to overcome difficulties and restrictions of the mesh-based methods, the alternative analysis methods have been proposed. We present a numerical technique for solving engineering analysis problems that combines meshfree method with distance fields, radial basis functions and collocation technique. The proposed approach enhances the collocation method with exact treatment of boundary conditions at all boundary points. It makes it possible to exclude boundary conditions from the collocation equations. This reduces the size of the algebraic system which results in faster solutions. On another hand, the boundary collocation points can be used to enforce the governing equation of the problem which enhances the solutions accuracy. Ability to use unstructured grids empowers the meshfree method with distance fields with higher level of geometric flexibility. In our presentation we demonstrate comparisons of the numerical results given by the combined approach with results delivered by the traditional collocation technique and meshfree method with distance fields.

Hybrid Method of Engineering Analysis: Combining Meshfree Method with Distance Fields and Collocation Technique

2011 ◽  
Vol 11 (3) ◽  
Author(s):  
Igor Tsukanov ◽  
Sudhir R. Posireddy
Keyword(s):  
Boundary Conditions ◽  
Unstructured Grids ◽  
Numerical Technique ◽  
Meshfree Method ◽  
Basis Functions ◽  
Benchmark Problems ◽  
Hybrid Technique ◽  
Engineering Analysis ◽  
Distance Fields ◽  
Collocation Technique

This paper describes a numerical technique for solving engineering analysis problems that combine radial basis functions and collocation technique with meshfree method with distance fields, also known as solution structure method. The proposed hybrid technique enables exact treatment of all prescribed boundary conditions at every point on the geometric boundary and can be efficiently implemented for both structured and unstructured grids of basis functions. Ability to use unstructured grids empowers the meshfree method with distance fields with higher level of geometric flexibility. By providing exact treatment of the boundary conditions, the new approach makes it possible to exclude boundary conditions from the collocation equations. This reduces the size of the algebraic system, which results in faster solutions. At the same time, the boundary collocation points can be used to enforce the governing equation of the problem, which enhances the solution’s accuracy. Application of the proposed method to solution of heat transfer problems is illustrated on a number of benchmark problems. Modeling results are compared with those obtained by the traditional collocation technique and meshfree method with distance fields.

scholarly journals Solving Transient Groundwater Inverse Problems Using Space–Time Collocation Trefftz Method

Water ◽  
2020 ◽  
Vol 12 (12) ◽  
pp. 3580
Author(s):  
Cheng-Yu Ku ◽  
Li-Dan Hong ◽  
Chih-Yu Liu
Keyword(s):  
Inverse Problems ◽  
Boundary Conditions ◽  
Numerical Solutions ◽  
Separation Of Variables ◽  
Time Integration ◽  
Meshfree Method ◽  
Space Time ◽  
Basis Functions ◽  
Trefftz Method ◽  
Dimensional Boundary

This paper presents a space–time meshfree method for solving transient inverse problems in subsurface flow. Based on the transient groundwater equation, we derived the Trefftz basis functions utilizing the method of separation of variables. Due to the basis functions completely satisfying the equation to be solved, collocation points are placed on the space–time boundaries. Numerical solutions are approximated based on the superposition theorem. Accordingly, the initial and boundary conditions are both regarded as space–time boundary conditions. Forward and inverse examples are conducted to validate the proposed approach. Emphasis is placed on the two-dimensional boundary detection problem in which the nonlinearity is solved using the fictitious time integration method. Results demonstrate that approximations with high accuracy are acquired in which the boundary data on the absent boundary may be efficiently recovered even when inaccessible partial data are provided.

Biharmonic navigation using radial basis functions

Robotica ◽  
2021 ◽  
pp. 1-12
Author(s):  
Xu-Qian Fan ◽  
Wenyong Gong
Keyword(s):  
Finite Difference ◽  
Boundary Conditions ◽  
Radial Basis Functions ◽  
Real World ◽  
Biharmonic Equation ◽  
Finite Difference Methods ◽  
Basis Functions ◽  
Large Size ◽  
Radial Basis ◽  
Difference Methods

Abstract Path planning has been widely investigated by many researchers and engineers for its extensive applications in the real world. In this paper, a biharmonic radial basis potential function (BRBPF) representation is proposed to construct navigation fields in 2D maps with obstacles, and it therefore can guide and design a path joining given start and goal positions with obstacle avoidance. We construct BRBPF by solving a biharmonic equation associated with distance-related boundary conditions using radial basis functions (RBFs). In this way, invalid gradients calculated by finite difference methods in large size grids can be preventable. Furthermore, paths constructed by BRBPF are smoother than paths constructed by harmonic potential functions and other methods, and plenty of experimental results demonstrate that the proposed method is valid and effective.

scholarly journals qEEG Analysis in the Diagnosis of Alzheimers Disease; a Comparison of Functional Connectivity and Spectral Analysis

2022 ◽  
Author(s):  
Maria Semeli Frangopoulou ◽  
Maryam Alimardani
Keyword(s):  
Spectral Analysis ◽  
Functional Connectivity ◽  
Data Science ◽  
Phase Synchronization ◽  
Connectivity Analysis ◽  
Alzheimers Disease ◽  
Brain Disorder ◽  
Alternative Analysis ◽  
Functional Connectivity Analysis ◽  
Analysis Methods

Alzheimers disease (AD) is a brain disorder that is mainly characterized by a progressive degeneration of neurons in the brain, causing a decline in cognitive abilities and difficulties in engaging in day-to-day activities. This study compares an FFT-based spectral analysis against a functional connectivity analysis based on phase synchronization, for finding known differences between AD patients and Healthy Control (HC) subjects. Both of these quantitative analysis methods were applied on a dataset comprising bipolar EEG montages values from 20 diagnosed AD patients and 20 age-matched HC subjects. Additionally, an attempt was made to localize the identified AD-induced brain activity effects in AD patients. The obtained results showed the advantage of the functional connectivity analysis method compared to a simple spectral analysis. Specifically, while spectral analysis could not find any significant differences between the AD and HC groups, the functional connectivity analysis showed statistically higher synchronization levels in the AD group in the lower frequency bands (delta and theta), suggesting that the AD patients brains are in a phase-locked state. Further comparison of functional connectivity between the homotopic regions confirmed that the traits of AD were localized in the centro-parietal and centro-temporal areas in the theta frequency band (4-8 Hz). The contribution of this study is that it applies a neural metric for Alzheimers detection from a data science perspective rather than from a neuroscience one. The study shows that the combination of bipolar derivations with phase synchronization yields similar results to comparable studies employing alternative analysis methods.

Free vibration analysis of rectangular and annular Mindlin plates with undamaged and damaged boundaries by the spectral collocation method

2011 ◽  
Vol 18 (11) ◽  
pp. 1722-1736 ◽  
Author(s):  
Ma’en S Sari ◽  
Eric A Butcher
Keyword(s):  
Free Vibration ◽  
Boundary Conditions ◽  
Collocation Method ◽  
Vibration Analysis ◽  
Natural Vibration ◽  
Numerical Technique ◽  
Free Vibration Analysis ◽  
Mindlin Plate ◽  
Grid Points ◽  
Matrix Vector

The objective of this paper is the development of a new numerical technique for the free vibration analysis of isotropic rectangular and annular Mindlin plates with damaged boundaries. For this purpose, the Chebyshev collocation method is applied to obtain the natural frequencies of Mindlin plates with damaged clamped boundary conditions, where the governing equations and boundary conditions are discretized by the presented method and put into matrix vector form. The damaged boundaries are represented by distributed translational and torsional springs. In the present study the boundary conditions are coupled with the governing equation to obtain the eigenvalue problem. Convergence studies are carried out to determine the sufficient number of grid points used. First, the results obtained for the undamaged plates are verified with previous results in the literature. Subsequently, the results obtained for the damaged Mindlin plate indicate the behavior of the natural vibration frequencies with respect to the severity of the damaged boundary. This analysis can lead to an efficient technique for structural health monitoring of structures in which joint or boundary damage plays a significant role in the dynamic characteristics. The results obtained from the Chebychev collocation solutions are seen to be in excellent agreement with those presented in the literature.

Solution of axisymmetric torsion problems by point matching

1971 ◽  
Vol 6 (2) ◽  
pp. 124-133 ◽  
Author(s):  
G J Matthews ◽  
C J Hooke
Keyword(s):  
Boundary Conditions ◽  
Numerical Technique ◽  
Point Matching ◽  
Analytical Results ◽  
Elasticity Equations ◽  
Matching Technique ◽  
Basic Solutions ◽  
Axisymmetric Bodies

A general numerical technique is presented for the solution of the problem of torsion of axisymmetric bodies. The method superimposes a number of basic solutions of the elasticity equations using the point-matching technique so as to satisfy approximately the prescribed boundary conditions of a body. Results obtained by this technique are compared with those obtained by alternative experimental and theoretical techniques for various body geometries to assess the accuracy of the method. The technique is then applied to the problem of the torsion of shouldered shafts since large discrepancies exist between the experimental and analytical results available for this type of structure.

A Solution of a Mixed Boundary Value Problem—Acoustic Pressure Fields in a Finite Container

1972 ◽  
Vol 94 (3) ◽  
pp. 643-648
Author(s):  
L. P. Solomon ◽  
N. Schryer
Keyword(s):  
Boundary Conditions ◽  
Pressure Field ◽  
Mixed Boundary ◽  
Numerical Technique ◽  
Boundary Value ◽  
Strong Function ◽  
Pressure Fields ◽  
End Conditions ◽  
Finite Domains

This paper investigates the effects of different boundary conditions in calculating pressure fields corresponding to incipient cavitation. We have utilized a technique which allows us to obtain a numerical solution of this problem for various frequencies and geometrical configurations. Our results provide evidence that determination of the pressure field is not only a function of depth but also a strong function of radius and whether or not the end conditions involve the use of a baffle. We have found that, particularly at the higher frequencies, the changing of the boundary conditions will cause large variations and differences in the pressure field. The numerical technique provides a method which allows the calculation of mixed boundary value problems associated with the reduced wave equation in finite domains. The technique specifies known error bounds. However, the distribution of errors over the domain is unknown.

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