Rational Mapping Function Research Articles

The stresses components in position and time of weakened plate with two holes conformally mapped into a unit circle by a conformal mapping with complex constant coefficients

<abstract> <p>In this paper, an infinite elastic plate weakened by two holes are considered and the complex variable method is used to derive a closed form of Gaursat functions for the first and second fundamental problems with variant time. The holes, in all previous works, are conformally mapped outside the unit circle without time. Here, the two holes are conformally mapped into the unit circle $ \varpi $ in the effect of time by the generalized rational mapping function with complex constant coefficients. By using this conformal mapping function, the fundamental problems transfer to an integro-differential equation with Cauchy kernel. Then, after applying the complex variable method, one can obtain a closed form of Gaursat functions. Some applications were discussed and the time effect on the applications was studied. In addition, the different stress components in each application have also been calculated using Maple 2022.1.</p> </abstract>

A New Mapped WENO Method for Hyperbolic Problems

In this study, a new family of rational mapping functions gRM(ω;k,m,s) is introduced for seventh-order WENO schemes. gRM is a more general family of mapping functions, which includes other mapping functions such as gM and gIM as special cases. The mapped WENO scheme WENO-IM(2,0.1), which uses gIM, performs excellently at fifth order but rather poorly at seventh order. The reason for this loss of accuracy was found to be the over-amplification of very small weights by the mapping process, which can be traced back to the large slope of gIM at ω = 0. For m > 1, gRM can be designed to have a unit slope at ω = 0, which will preserve small weights with little to no amplification. It has been demonstrated through several one-dimensional linear advection test cases that the mapped WENO scheme WENO-RM(6,3,2 × 103), which uses the mapping function gRM(ω;6,3,2 × 103), outperforms both WENO-M and WENO-IM(2,0.1) at seventh order. The proposed scheme also performs better at a number of one-dimensional inviscid gas flow problems compared to other popular WENO schemes such as the WENO-Z scheme.

Analytical solution of an orthotropic elasticity for a frictional rigid flat punch on a half plane with an oblique edge crack using a mapping function

A solution of an orthotropic elastic plane problem is derived for a frictional rigid flat punch on a half plane with an oblique edge crack. A mapping function is used for arbitrary configurations. No stress function using a mapping function seems to have been derived. The problem is solved as a Riemann–Hilbert problem. The final exact stress functions are represented by an irrational mapping function as a closed form. Stress components are represented by one complex variable. Arbitrarily shaped half plane problems can be solved by exchanging the mapping function. Therefore, once the analytical solution has been derived, calculating the stress components is simpler than for an isotropic problem. It is easier to use an irrational mapping function than to form a rational mapping function for an isotropic problem. As an example, the stress distributions are shown for a frictional coefficient and an oblique edge crack angle. Four elastic constants are expressed by two independent parameters.

On the PIV/PTV uncertainty related to calibration of camera systems with refractive surfaces

This paper investigates the calibration and measurement uncertainty related to the use of different camera models in optical systems that include refractive surfaces. A refractive surface is an interface between media with different optical properties which introduces distortions in the imaging process due to the refraction of the lines-of-sight. This is an issue common to all the investigations of fluids flowing around or inside transparent solid geometries and is of relevance for a strong curvature of the solid/fluid interface. Appropriate modelling of the refractive effects is possible by integrating the pinhole camera model with a ray-tracing method, as demonstrated in a previous work (Paolillo and Astarita 2020 IEEE Trans. Pattern Anal. Mach. Intell.). On the other side, analytical camera models with a pure mathematical foundation, like those based on polynomials or rational functions, are classically used in the PIV/PTV community. Due to the non-linear nature of the involved distortions, the accuracy of these models in representing the imaging process in presence of refractive geometries depends strongly on the polynomial order and noise of the data used for the calibration. The current work provides a numerical estimate of the uncertainty inherent to the analytical camera models by using data generated via a reference refractive camera model. The present results show that high accuracy requires high orders, which implies a large number of calibration parameters and high demand for computational resources. In particular, the rational mapping functions exhibit superior performance compared to the polynomials, although their calibration is found to be sensitive to image noise and they might yield large extrapolation errors. An experimental verification is also reported, which shows that for the estimation of the velocity statistics a 7th order polynomial model offers results comparable to those of a refractive camera model.

Analysis of a mixed boundary value problem for an orthotropic elasticity using a mapping function

A solution is derived for a mixed boundary orthotropic elastic plane problem using a mapping function to solve arbitrary configurations. No stress function using a mapping function seems to have been derived. The problem is solved as a Riemann–Hilbert problem. The final exact stress functions are represented by an irrational mapping function as a closed form. Stress components are represented by one complex variable. Arbitrarily shaped hole problems can be solved by changing the mapping function. Once the analytical solution has been derived, calculating the stress components is simpler than for an isotropic problem. It is easier to use an irrational mapping function than to form a rational mapping function for an isotropic problem. As an example, an infinite plane with a square hole subjected to uniform tension is analyzed. The stress distributions are shown for Cases I and III problems corresponding to two characteristic roots of the fundamental equation. In a Case III problem, the symmetry of stress distributions is lost. A solution of an external force boundary value problem can be derived from that of the mixed boundary value problem.

Bimaterial problems with an interfacial cavity for some boundary conditions subjected to uniform heat flux normal to the interface

Stress analysis is carried out for a bimaterial infinite plane with an interfacial cavity. Uniform heat flux applies to the normal to the interface. Four combinations of boundary conditions are considered, that is, isothermal and adiabatic boundary conditions for heat flux analysis, and external force and displacement boundary conditions for stress analyses. The infinite plane consists of two bonded dissimilar materials of a half plane with a single notch. To achieve analytical solutions, a rational mapping function and a complex variable method are used. By changing the mapping function, other geometries for the notch can be analyzed. Complex stress functions for isothermal and external boundary conditions can be only achieved for stress calculation. The stress intensities of debonding are investigated for various debonding lengths for some elliptical holes, and for the debonding extensions. Complex stress functions for isothermal and displacement boundary conditions can be expressed by an infinite series and stress components et al. cannot be calculated. However, a solution of interfacial rigid inclusion can be solved. Complex stress functions for the adiabatic boundary are achieved by the integral forms for external force and displacement boundary conditions, and the integral cannot be carried out, and therefore, stress components cannot be achieved.

Bimaterial infinite plane with a cavity on the interface subjected to different temperatures

Stress analysis is applied to two bonded materials of infinite extent with a single notch at their interface. The two materials, represented in normal view as two joined half planes with a notch, are subjected to different temperatures. As illustration, a bimaterial infinite plane with an elliptical hole at the interface is analyzed. To achieve a closed-form solution, a rational mapping function and a complex variable method are used. By changing the mapping function, other geometries for the notch can be analyzed. The cavity at the interface constitutes a small defect because its extent is small compared with the infinite plane. A mathematical difficulty arises with the presence of the points at infinity of the half planes. The coefficients of the homogeneous part of the stress function are expressible in terms of Dundurs’ parameters, but the temperature-dependent loading term is not. Stress distributions without and with debonding for two pairings of bimaterial are shown. The stress intensity of debonding is defined, and values are investigated for various debonding lengths for the elliptical holes. Also the debonding extension is investigated. As a special case, the stress intensity factors for T-shaped cracks are also investigated.

Stress intensity factors of a rhombic hole with symmetric cracks under uniform transverse thin plate bending

A thin plate bending problem (Kirchhoff thin plate) is solved analytically for an infinite plate with a rhombic hole with symmetric cracks at the both corners. The loading is a uniform transverse bending at infinity. The stress distribution and the stress intensity factors of the bending are obtained for a range of corner angles and crack lengths of the rhombic hole. Approximate expressions of the stress intensity factor are proposed for some corner angles and Poisson’s ratios using the stress distribution expressed by the roots of the characteristic equation of the sharp notched plate before the crack initiation. The precision of the expressions are investigated. Characteristic factors for the approximate expressions are stated. A rational mapping function of a sum of fractional expressions and complex stress functions are used for the stress analysis.

A New Mapped Weighted Essentially Non-oscillatory Method Using Rational Mapping Function

In this paper, we develop a new mapped weighted essentially non-oscillatory method for hyperbolic conservation laws by proposing a new family of mapping functions, which includes the mapping functi...

Solution of a bonded bimaterial problem of two interfaces subjected to different temperatures

A closed-form solution is derived for the bonded bimaterial planes at two interfaces subjected to different temperatures. The bonded planes with two interfaces are symmetric with respect to the interface, which is straight. A rational mapping function and complex stress functions are used for the analysis. The problem is reduced to a Riemann–Hilbert problem. The solution includes an integral term. This integral cannot be carried out. However, the first derivative of complex stress functions which does not include integral terms with regard to the variables of the mapping plane is achieved. Therefore, there is no need for numerical integration to calculate stress components and to determine unknown coefficients in a complex stress function. This is very beneficial. It is more difficult to derive the solution to the two-interface problem compared to the general solution. As a demonstration of geometry, semi-strips bonded in two places at the ends of strips are considered. Unbonded parts are a model of debonding. The solution for different geometrical shapes can be obtained by changing the mapping function. Some stress distributions are shown for different lengths of the interface. The stress intensity of debonding (SID) (corresponding to the root of strain energy release rate) is investigated for the debonding extension. SID is the same as the root of strain energy release rate for the evaluation of the strength at the debonding tip.

Solution of bonded bimaterial problem of two interfaces subjected to concentrated forces and couples

A closed form solution is derived for the bonded bimaterial planes at two interfaces. The bonded planes with two interfaces are symmetric with respect to the interface, which is straight. A rational mapping function and complex stress functions are used for the analysis. The problem is reduced to a Riemann‐Hilbert problem. Two interfaces problem to derive the general solution is more difficult than one interface problem. As a demonstration of geometry, semi‐strips bonded at two parts at the ends of strips are considered. The solution of different geometrical shapes can be obtained by changing the mapping function. Concentrated forces and couples are applied to the each strip. The first derivative of complex stress functions which does not include integral terms with regard to variable of the mapping plane is achieved. Therefore, there is no need of numerical integration to calculate stress components and to determine unknown coefficients in complex stress function. This is very benefit. All elastic constants in complex stress functions are expressed by Dundurs’ parameters. Stress distributions are shown for different lengths of the interface.

On a Problem of an Infinite Plate with a Curvilinear Hole inside the Unit Circle

In this work, we used the complex variable methods to derive the Goursat functions for the first and second fundamental problem of an infinite plate with a curvilinear hole C. The hole is mapped in the domain inside a unit circle by means of the rational mapping function. Many special cases are discussed and established of these functions. Also, many applications and examples are considered. The results indicate that the infinite plate with a curvilinear hole inside the unit circle is very pronounced.

A crack initiation and two debondings development at the interface of a circular rigid inclusion under uniform loading

Competition of a crack and two debondings at the interface of a circular rigid inclusion in an infinite elastic body is analyzed under uniaxial loading in the x and y directions, respectively, and under biaxial uniform loading. It is investigated how these debondings develop along the interface of the inclusion from the initial debondings and where a crack occurs from the tip of debonding. Particularly, when there are both possibilities of the debonding development and of the crack occurrence from the tip of the debonding, it can be decided which phenomenon actually occurs. The angles at which the debondings develop and the crack occurs are determined. The magnitude of the load is also determined. As the criterion for debonding development and crack occurrence, strain energy release rates are used. Moreover, the restricting condition is that the normal stress at the interface ahead of the debonding tip is positive and the stress intensity factor of mode I just after crack occurrence is positive. As for the load, the constant load and the gradually increasing load from zero are considered. The stress analysis is carried out as a mixed boundary value problem with two stress and two displacement boundaries of plane elasticity. The rational mapping function of a sum of fractional expressions and complex stress functions are used for the stress analysis.

Mixed boundary value problem for quasi-orthotropic elastic plane

The quasi-orthotropic elastic plane in which the characteristic roots of the fundamental differential equation for the orthotropic elastic plane are doubled is investigated for a mixed boundary value problem. One of the main purposes of the present paper is to derive a general solution of a mixed boundary value problem. For the associated stress analysis, a rational mapping function is used and a closed-form stress function is obtained. The stress analysis is hence rigorous for the rational mapping function, which gives a comparative arbitrary configuration. The quasi-orthotropic elastic plane problem includes that of an isotropic elastic plane. As a demonstration of the stress analysis, a half plane with an oblique edge rigid line inclusion in which the rim of the half plane and the principal axis for the orthotropic elasticity coincides is analyzed. A uniform tension is applied over the half plane, and the rigid line inclusion is expressed as a fixed boundary. The homogenous solution of the stress function can be obtained without any integration. Stress distributions and stress intensity factors at the tip of the rigid line inclusion are investigated for Poisson’s ratio. The relationship between the quasi-orthotropic elastic plane and the isotropic elastic plane with respect to stress intensity factors is derived. Stress functions in which the rim of the half plane and the principal axis of the orthotropic elasticity do not coincide are also described.

Designing multiband coupled-resonator filters using reactance transformations

A method of constructing multiband filtering functions from single-band functions using rational reactance mapping functions is presented. The technique can be applied to coupled-resonator filters with N pass-bands, each of arbitrary bandwidth and centre frequency. The resulting filter topologies are simple to implement and do not introduce cross-couplings beyond that of the original low-pass prototype. © 2014 Wiley Periodicals, Inc. Int J RF and Microwave CAE 25:81–92, 2015.

An arbitrarily shaped inclusion with uniform eigencurvatures in an infinite plate, semi-infinite plate, two bonded semi-infinite plates or a circular plate

Within the framework of the Kirchhoff–Love isotropic and homogeneous plate theory, we obtain, in a unified manner, the analytic solutions to the Eshelby’s problem of an inclusion of arbitrary shape with uniform eigencurvatures in an infinite plate, a semi-infinite plate, one of two bonded semi-infinite plates or a circular plate by means of conformal mapping and analytical continuation. The edge of the semi-infinite plate can be rigidly clamped, free or simply supported, while that of the circular plate can be rigidly clamped, free or perfectly bonded to the surrounding infinite plate. Several examples of practical and theoretical interests are presented to demonstrate the general method. In particular, the elementary expressions of the internal elastic fields of bending moments and deflections within an (n + 1)-fold rotational symmetric inclusion described by a five-term mapping function, a symmetric airfoil cusp inclusion, a symmetric lip cusp inclusion and an inclusion described by a rational mapping function in an infinite plate are derived.

Solution of problem of two dissimilar materials bonded at one interface subjected to temperature

A solution of problem of two dissimilar materials bonded at one interface subjected to temperature is derived. To obtain a closed-form solution, a rational mapping function and a complex variable method are used. The coefficients of the homogeneous part of the stress function are expressed by Dunders’ parameters, but loading term of temperature is not expressed by them. As a demonstration, semi-strips bonded at one part at the ends are considered. The each strip is subjected to uniform temperature. Examples of stress distributions are shown. The relations of stress and temperature on the interface are described. Debondings on both sides of the interface are considered. Stress intensity of debonding (SID) is defined, and the values are investigated for various debonding lengths. And the debonding extension or the crack initiation into the material is investigated. The effects of material constants (Dundurs’ parameter) on SID are also investigated. By changing mapping function, other geometries can be analyzed.

Competition of crack and debonding at the interface of a circular rigid inclusion under uniform loading

Competition of a crack and a debonding at the interface of a circular rigid inclusion in an infinite elastic body is analyzed under uniaxial loading in the x and y directions, respectively, and under biaxial uniform loading. It is investigated how the debonding develops along the interface of the inclusion from the initial debonding and where the debonding stops and a crack occurs from the tip of debonding. Particularly when there are both possibilities of the debonding development and of the crack occurrence from the tip of the debonding, it can be decided which phenomenon actually occurs. The angles at which the debonding develops and the crack occurs are determined. As the criterion for debonding development and crack occurrence at the debonding tip, strain energy release rates are used. Moreover, the restricting condition is that the normal stress at the interface ahead of the debonding tip is positive and the Mode I stress intensity factor just after crack occurrence is positive. As the loading, the constant load and the gradually increasing load from zero are considered. The stress analysis is carried out as a mixed boundary value problem of plane elasticity. As the stress analysis, the rational mapping function of a sum of fractional expressions and complex stress functions are used and closed form stress functions are derived.

An Infinite Elastic Plate Weakened by a Generalized Curvilinear Hole and Goursat Functions

Complex variables method has been used to solve the first and second fundamental problems for an infinite plate weakened by a generalized curvilinear hole C. The curvilinear hole is conformally mapped on the domain outside or inside a unit circle γ using a general rational mapping function with complex constants. Many special and new cases are derived from this work. Some of the work of the previous authors in this domain will be considered as special cases of this paper. Also the interesting cases when the shape of the hole takes different famous shapes are included. The components of stresses for some examples are obtained.

Stress analysis of quasi-orthotropic elastic plane

The quasi-orthotropic elastic plane in which the characteristic roots of the fundamental differential equation for the orthotropic elastic plane are doubled is investigated. For the associated stress analysis, a rational mapping function is used and a closed-form solution is obtained. Therefore, the stress analysis is rigorous for the rational mapping function. The stress functions can be obtained without any integration. As a demonstration of the stress analysis, a half plane with an oblique edge crack is analyzed. Stress distributions and stress intensity factors are investigated. The relationships between the quasi-orthotropic elastic plane and isotropic plane with respect to the stress intensity factor are investigated. It is also determined that the stress intensity factors for the quasi-orthotropic elastic plane can be calculated from those of the isotropic elastic plane using a similar configuration and loading condition.