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Partial Differential Equations Commons™
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- Inverse problems (12)
- Impedance imaging (5)
- Inverse Problem (4)
- Non-destructive testing (4)
- Corrosion (3)
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- Nondestructive testing (3)
- Thermal imaging (3)
- Cloaking (2)
- Heat Equation (2)
- Thermal Imaging (2)
- Thermal resistance (2)
- Thermal testing (2)
- Crack detection (1)
- Difference Equations (1)
- Eigenvalues (1)
- Eigenvectors (1)
- Electrical Impedance (1)
- Electrical Impedance Tomography (1)
- Hamiltonian Systems (1)
- Imaging (1)
- Imnerse problems (1)
- Impedance Imaging (1)
- Invisibility (1)
- Jump condition. (1)
- Linearization (1)
- Maximum Entropy Methods (1)
- Non-destructive (1)
- Non-destructive thermal testing (1)
- Nonlinear boundary condition (1)
- Numerical Methods (1)
Articles 1 - 24 of 24
Full-Text Articles in Partial Differential Equations
Thermal Detection Of Inaccessible Corrosion, Matthew Charnley, Andrew Rzeznik
Thermal Detection Of Inaccessible Corrosion, Matthew Charnley, Andrew Rzeznik
Mathematical Sciences Technical Reports (MSTR)
In this paper, we explore the mathematical inverse problem of detecting corroded material on the reverse side of a partially accessible metal plate. We will show how a linearization can be used to simplify the initial problem and explain a regularization method used to obtain acceptable results for the corrosion profile. We will also state and perform some calculations for the full three-dimensional problem for possible future work.
Nondestructive Electrothermal Detection Of Corrosion, Brittany Ambeau, Harris Enniss, Stefan Schnake
Nondestructive Electrothermal Detection Of Corrosion, Brittany Ambeau, Harris Enniss, Stefan Schnake
Mathematical Sciences Technical Reports (MSTR)
Nondestructive testing and imaging plays an important role in many industries, e.g., the monitoring and maintenance of corrosion in aircraft. The general technique is to input energy in some form into an object, observe the object’s response, and from this input-output information determine the internal structure. New techniques are always being explored, and recently there has been much interest in methods that use multiple forms of energy. In this vein, we examine a new technique for imaging corrosion or material loss in an object by combining electrical and thermal measurements on some accessible portion of the object’s outer boundary. The …
Cloaking Against Thermal Imaging, Maple So
Cloaking Against Thermal Imaging, Maple So
Mathematical Sciences Technical Reports (MSTR)
There has been a lot of recent interest in cloaking and invisibility in the mathematics and science communities, and in fact physically plausible mechanisms have been proposed (some built) for cloaking an object against detection using a variety of electromagnetic methods. The ideas are very general, however, and should allow one to design cloaks that work against other forms of imaging. We examine the possibility of cloaking an object to make it invisible to an observer using thermal energy (heat) as the imaging tool. Specifically, we desire to cloak an object inside a two-dimensional disk by cutting a small hole …
Electrical Impedance Imaging Of Corrosion On A Partially Accessible 2-Dimensional Region, Court Hoang, Katherine Osenbach
Electrical Impedance Imaging Of Corrosion On A Partially Accessible 2-Dimensional Region, Court Hoang, Katherine Osenbach
Mathematical Sciences Technical Reports (MSTR)
In this paper we examine the inverse problem of determining the amount of corrosion on an inaccessible surface of a two-dimensional region. Using numerical methods, we develop an algorithm for approximating corrosion profile using measurements of electrical potential along the accessible portion of the region. We also evaluate the effect of error on the problem, address the issue of ill-posedness, and develop a method of regularization to correct for this error. An examination of solution uniqueness is also presented.
Anti-Cloaking: The Mathematics Of Disguise, Theresa C. Anderson, Brooke E. Phillips
Anti-Cloaking: The Mathematics Of Disguise, Theresa C. Anderson, Brooke E. Phillips
Mathematical Sciences Technical Reports (MSTR)
Recent developments in cloaking, the ability to selectively bend electromagnetic waves so as to render an object invisible, have been abundant. Based on cloaking principles, we will describe several ways to mathematically disguise objects in the context of electrical impedance imaging. Through the use of a change-of-variables scheme we show how one can make an object appear enlarged, translated, or rotated by surrounding it with a suitable "metamaterial," a man-made material that selectively redirects current. Analysis of eigenvectors and eigenvalues, which describe how current flows, follow. We prove that in order to disguise an object, a metamaterial must encompass both …
Determining The Shape Of A Resistor Grid, Esther Chiew, Vincent Selhorst-Jones
Determining The Shape Of A Resistor Grid, Esther Chiew, Vincent Selhorst-Jones
Mathematical Sciences Technical Reports (MSTR)
Impedance imaging has received a lot of attention in the past two decades, as a means for non-destructively imaging the interior of a conductive object. One injects a known electrical current pattern into an object at the exterior boundary, then measures the induced potential (voltage) on some portion of the boundary. The goal is to recover information about the interior conductivity of the object, which (we hope) influences the voltages we measure. Of course one can also use multiple input currents and measured voltages. A variation on this problem is that of "boundary identification," in which some portion of the …
Utilizing Thermal Testing For Recovering, James Preciado, Thomas Werne
Utilizing Thermal Testing For Recovering, James Preciado, Thomas Werne
Mathematical Sciences Technical Reports (MSTR)
Given a two-dimensional region that contains one or more circular voids, we develop mathematical methods to locate the center and radius of the voids based on thermal boundary data. These methods can be readily applied in the field of non-destructive evaluation.
Thermal Imaging Of Circular Inclusions Within A Two-Dimensional Region, Shannon Talbott, Hilary Spring
Thermal Imaging Of Circular Inclusions Within A Two-Dimensional Region, Shannon Talbott, Hilary Spring
Mathematical Sciences Technical Reports (MSTR)
The ability to study the interior of an object without destroying it is an important industrial tool. One method of recent interest is steady-state thermal or impedance imaging. In this paper we will use the steady-state heat equation to locate one or more circular inclusions within a two-dimensional region, where the boundaries of the inclusions have partially disbonded from the surrounding material; this disbond is modelled as between the heat flux and jump discontinuity at the disbonded interface.
Time-Dependent Thermal Imaging Of Circular Inclusions, Donald L. Brouwn, Mark Hubenthal
Time-Dependent Thermal Imaging Of Circular Inclusions, Donald L. Brouwn, Mark Hubenthal
Mathematical Sciences Technical Reports (MSTR)
This paper considers the inverse problem of locating one or more circular inclusions in a two-dimensional domain using thermal boundary data, specifically, the input heat flux and measured boundary temperature. The forward problem is governed by the heat equation. We show how the position and size of such defects can be recovered using the boundary data and various approximations of the solution to the forward problem. We also consider the stability of the algorithm involved to recover the defects.
Reconstruction Of Partially Conductive Cracks Using Boundary Data, David Mccune, Janine Haugh
Reconstruction Of Partially Conductive Cracks Using Boundary Data, David Mccune, Janine Haugh
Mathematical Sciences Technical Reports (MSTR)
This paper develops an algorithm for finding one or more non-insulated, pair-wise disjoint, linear cracks in a two dimensional region using boundary measurements.
Non-Destructive Testing Of Thermal Resistances For A Single Inclusion In A 2-Dimensional Domain, Nicholas Christian, Mathew A. Johnson
Non-Destructive Testing Of Thermal Resistances For A Single Inclusion In A 2-Dimensional Domain, Nicholas Christian, Mathew A. Johnson
Mathematical Sciences Technical Reports (MSTR)
In this paper we examine the inverse problem of determining the amount of corrosion/disbonding which has occurred on the boundary of a single circular (or nearly circular) inclusion D in a two dimensional domain W using Cauchy data for the steady-state heat equation. We develop an algorithm for reconsructing a function which qunatifies the level of corrosion/disbonding at each point in ¶W. We also address the issue of well-posedness and develop a simple regularization scheme. Then we provide several numerical examples. We shall show a simple procedure for recovering the center of D assuming that the boundary of W and …
Reconstruction Of An Unknown Boundary Portion From Cauchy Data In N-Dimensions, Kurt M. Bryan, Lester Caudill
Reconstruction Of An Unknown Boundary Portion From Cauchy Data In N-Dimensions, Kurt M. Bryan, Lester Caudill
Mathematical Sciences Technical Reports (MSTR)
We consider the inverse problem of determining the shape of some inacces sible portion of the boundary of a region in n dimensions from Cauchy data for the heat equation on an accessible portion of the boundary. The inverse problem is quite ill-posed, and nonlinear. We develop a Newton-like algorithm for solving the problem, with a simple and efficient means for computing the required derivatives, develop methods for regularizing the process, and provide computational examples
Determining The Length Of A One-Dimensional Bar, Natalya Yarlikina, Holly Walrath
Determining The Length Of A One-Dimensional Bar, Natalya Yarlikina, Holly Walrath
Mathematical Sciences Technical Reports (MSTR)
In this paper we examine the inverse problem of determining the length of a one-dimensional bar from thermal measurements (temperature and heat flux) at one end of the bar (the "accessible" end); the other inaccessible end of the bar is assumed to be moving. We develop two different approaches to estimating the length of the bar, and show how one approach can also be adapted to find unknown boundary conditions at the inaccessible end of the bar.
A Review Of Selected Works On Crack Indentification, Kurt M. Bryan
A Review Of Selected Works On Crack Indentification, Kurt M. Bryan
Mathematical Sciences Technical Reports (MSTR)
We give a short survey of some of the results obtained within the last 10 years or so concerning crack identification using impedance imaging techniques. We touch upon uniqueness results, continuous dependence results, and computational algorithms.
Characterizing A Defect In A One-Dimensional Bar, Cynthia Gangi, Sameer Shah
Characterizing A Defect In A One-Dimensional Bar, Cynthia Gangi, Sameer Shah
Mathematical Sciences Technical Reports (MSTR)
We examine the inverse problem of locating and describing an internal point defect in a one dimensional rod W by controlling the heat inputs and measuring the subsequent temperatures at the boundary of W. We use a variation of the forward heat equation to model heat flow through W, then propose algorithms for locating an internal defect and quantifying the effect the defect has on the heat flow. We implement these algorithms, analyze the stability of the procedures, and provide several computational examples.
Reconstruction Of Cracks With Unknown Transmission Condition From Boundary Data, F Ronald Ogborne Iii, Melissa E. Vellela
Reconstruction Of Cracks With Unknown Transmission Condition From Boundary Data, F Ronald Ogborne Iii, Melissa E. Vellela
Mathematical Sciences Technical Reports (MSTR)
We examine the problem of Identifying both the location and constitutive law governing electrical current flow across a one-dimensional linear crack in a two dimensional region when the crack only partially blocks the flow of current. We develop a a constructive numerical procedure for solving the inverse problem and provide computational examples.
Fast Reconstruction Of Cracks Using Boundary Measurements, Nicholas A. Trainor, Rachel M. Krieger
Fast Reconstruction Of Cracks Using Boundary Measurements, Nicholas A. Trainor, Rachel M. Krieger
Mathematical Sciences Technical Reports (MSTR)
This paper develops a fast algorithm for locating one or more perfectly insulating, pair-wise disjoint, linear cracks in a homogeneous two-dimensional electrical conductor, using boundary measurements.
Solvability Of A Parabolic Boundary Value Problem With Internal Jump Condition, Kurt M. Bryan, Lester Caudill
Solvability Of A Parabolic Boundary Value Problem With Internal Jump Condition, Kurt M. Bryan, Lester Caudill
Mathematical Sciences Technical Reports (MSTR)
We examine a model for the propagation of heat through a one-dimensional object with an interior ''flaw". The flaw is modeled as a nonlinear relationship between the flux and temperature jump at an interior point of the object. Under realistic hypotheses, the resulting nonlinear initial boundary value problem is shown to have a unique and suitably smooth solution.
Stability And Reconstruction For An Inverse Problem For The Heat Equations, Kurt M. Bryan, Lester Caudill
Stability And Reconstruction For An Inverse Problem For The Heat Equations, Kurt M. Bryan, Lester Caudill
Mathematical Sciences Technical Reports (MSTR)
We examine the inverse problem of determining the shape of some unknown portion of the boundary of a region W from measurements of the Cauchy data for solutions to the heat equation on W. By suitably linearizing the inverse problem we obtain uniqueness and continuous dependence results. We propose an algorithm for recovering estimates of the unknown portion of the surface and use the insight gained from a detailed analysis of the inverse problem to regularize the inversion. Several computational examples are presented.
Uniqueness For A Boundary Identification Problem In Thermal Imaging, Kurt M. Bryan, Lester F. Caudill
Uniqueness For A Boundary Identification Problem In Thermal Imaging, Kurt M. Bryan, Lester F. Caudill
Mathematical Sciences Technical Reports (MSTR)
An inverse problem for a parabolic initial-boundary value problem is considered. The goal is to determine an unknown portion of the boundary of a region in Rn from measurements of Dirichlet data on a known portion of the boundary. It is shown that under reasonable hypotheses uniqueness results hold.
Effective Behavior Of Clusters Of Microscopic Cracks Inside A Homogeneous Conductor, Kurt M. Bryan, Michael Vogelius
Effective Behavior Of Clusters Of Microscopic Cracks Inside A Homogeneous Conductor, Kurt M. Bryan, Michael Vogelius
Mathematical Sciences Technical Reports (MSTR)
We study the effective behaviour of a periodic array of microscopic cracks inside a homogeneous conductor. Special emphasis is placed on a rigorous study of the case in which the corresponding effective conductivity becomes nearly singular, due to the fact that adjacent cracks nearly touch. It is heuristically shown how thin clusters of such extremely close cracks may macroscopically appear as a single crack. The results have implications for our earlier work on impedance imaging.
An Inverse Problem In Thermal Language, Kurt M. Bryan, Lester Caudill
An Inverse Problem In Thermal Language, Kurt M. Bryan, Lester Caudill
Mathematical Sciences Technical Reports (MSTR)
This paper examines uniqueness and stability results for an inverse problem in thermal imaging. The goal is to identify an unknown boundary of an object by applying a heat flux and measuring of the induced temperature on the boundary of the sample. The problem is studied both in the case in which one has of data at every point on the boundary of the region and the case in which only finitely many measurements are available. An inversion procedure is developed and used to study the stability of the inverse problem for various experimental configurations.
Reconstruction Of Multiple Cracks From Experimental, Electrostatic Boundary Measurements, Kurt M. Bryan, Valdis Liepa, Michael Vogelius
Reconstruction Of Multiple Cracks From Experimental, Electrostatic Boundary Measurements, Kurt M. Bryan, Valdis Liepa, Michael Vogelius
Mathematical Sciences Technical Reports (MSTR)
We demonstrate the viability of using Electrical Impedance Tomography (EIT) for the reconstruction of multiple macroscopic cracks in a conductive medium.
Time-Discretization Of Hamiltonian Dynamical Systems, Yosi Shibberu
Time-Discretization Of Hamiltonian Dynamical Systems, Yosi Shibberu
Mathematical Sciences Technical Reports (MSTR)
Difference equations for Hamiltonian systems are derived from a discrete variational principle. The difference equations completely determine piecewise-linear, continuous trajectories which exactly conserve the Hamiltonian function at the midpoints of each linear segment. A generating function exists for transformations between the vertices of the trajectories. Existence and uniqueness results are present as well as simulation results for a simple pendulum and an inverse square law system.