Reinhard Piltner | Georgia Southern University (original) (raw)

Papers by Reinhard Piltner

For several problems in mechanics the solution for the displacements, strains and stresses can be... more For several problems in mechanics the solution for the displacements, strains and stresses can be expressed in terms of complex valued functions. With the use of the complex valued functions the satisfaction of the governing differential equations is guaranteed a priori. Here different types of approximation functions are derived using discretized Cauchy integrals. The boundary of the Cauchy integrals can be the boundary of the solution domain or the boundary of a finite element domain. The computation of the real approximation functions for the displacements and stresses involve complex logarithmic terms which are multiplied by powers of the complex variable z=x+iy. The constructed approximation functions are used for finite element computations and for simulations involving only the boundary of the solution domain. For a variety of numerical examples high accuracy for displacements and stresses could be obtained

Local Effects in the Analysis of Structures, 1985

Abstract Using two different variational formulations some special finite elements as for example... more Abstract Using two different variational formulations some special finite elements as for example elements with circular and elliptic holes, with internal and external cracks have been developed and tested numerically. These elements are problem adapted as the trial functions satisfy not only the governing differential equations but also some boundary conditions on such influential boundary portions as crack or hole surfaces. So the local character of a solution is taken into account appropriately. Even problems with singularities as crack and sharp corner problems for example can be treated numerically in an effective manner. In both methods used one obtains symmetric element matrices which can be evaluated via simple numerical integrations along the element boundaries. The special problem-adapted finite elements can be coupled with conventional elements. The discussion of the special elements includes the variational formulations and element coupling procedures as well as the construction of special trial functions for 2- and 3-dimensional elasticity problems.

Lecture Notes in Computer Science, 2004

Journal of Environmental Informatics, 2008

It is important to conduct research on the connection between air pollution and human health usin... more It is important to conduct research on the connection between air pollution and human health using population-based spatiotemporal environmental exposure assessment on a large scale with respect to area and population. In our paper, using a set of spatiotemporal data with annual ozone concentration measurements in the contiguous U.S. during 1994 and 1999, we address the following challenging issues in conducting such research: spatiotemporal interpolation, comparison of spatiotemporal interpolation methods, visualization, and analysis of population exposure to ozone. A 3D shape function based spatiotemporal interpolation method has been used in this paper to estimate the ozone concentrations at any unmeasured location and time. Using the leave-one-out cross-validation, we compute error statistics to compare the shape function and IDW (Inverse Distance Weighting) methods. For the considered case studies it is observed that the shape function method is better than IDW in terms of MAPE (Mean Absolute Percentage Error) and algorithm complexity. For generating maps of annual ozone concentrations, we propose a new approach to select locations to interpolate and visualize: picking U.S. census block centroids as sample locations. The advantage of this approach is to generate more sample points in areas with more intensive human activities. In our experiment, there were about 8,000,000 sample points selected per year. Traditional GIS techniques are insufficient in handling such kind of spatiotemporal data. The visualization results of ozone concentration distribution at the census tract level in the contiguous U.S. from 1994 to 1999 are illustrated. We also analyze the population exposure to ozone in the year 1999 according to different ozone concentration levels following the recommendations given by the U.S. EPA on air quality. Our finding is that in the year of 1999, 9.8% total population in the contiguous U.S. has been exposed to a high risk ozone level, 78.7% to a moderate risk, and only 11.5% to a low risk.

Spatiotemporal Analysis of Air Pollution and Its Application in Public Health

Abstract Air pollution has been a major risk to public health. It is imperative to monitor the sp... more Abstract Air pollution has been a major risk to public health. It is imperative to monitor the spatiotemporal patterns of regional air pollution. However, air pollution data are often collected at a limited set of locations and archived at different times. In order to monitor pollution in the continuous space-time domain, traditional interpolation methods tend to treat space and time separately when estimating the pollution data at un-sampled locations and times. But such methods may not be able to yield satisfactory interpolation results. In addition, a vast amount of air quality data have been available to request in real time with the advance of modern sensors, but existing approaches are limited in their ability to process large data sets and support real-time visualizations. In this research, we investigate and compare several spatiotemporal interpolation methods with the goal to conduct interpolation on real-time air pollution data at a large geographic area. Both accuracy and efficiency are evaluated in this study. Based on the findings, we developed a visualization approach using a proposed method that allows real-time summarization and presentation of hourly air pollution data across the contiguous United States. A web application is developed that provides a portal to the public to visualize air quality.

The increasing amount of air pollution data requires efficient spatiotemporal interpolation metho... more The increasing amount of air pollution data requires efficient spatiotemporal interpolation methods to handle the demanding computational tasks. Several interpolation methods have been explored in the previous studies. Although many geographic information system applications provide interpolation tools, most current interpolation methods only apply to spatial data. Air pollution data not only have spatial attributes, but also change with time. When interpolating across space and time, the choice of the time scale versus the distance scale is an important issue that affects the accuracy of interpolation. In this chapter, we introduced some efficient deterministic spatiotemporal interpolation methods and search for optimal parameters to achieve good accuracy.

Mixed-enhanced quadrilateral finite elements with Wachspress-type functions. Simo and Rifai intro... more Mixed-enhanced quadrilateral finite elements with Wachspress-type functions. Simo and Rifai introduced the method of "enhanced strains" in 1990. Since then the method became quite popular to improve the performance of low order finite elements. The method has been used for both linear and non-linear problems by several researchers. Piltner and Taylor discussed an alternative to the original enhanced strain concept. In the alternative version of the enhanced strain method, a modified Hu-Washizu variational formulation is used. Stresses, strains, and enhanced strains are assumed in addition to the displacements. Previously, bilinear displacement shape functions assumed in a mapped bi-unit square have been used. For the current research Wachspress-type rational functions are utilized for the displacements. A special feature of the Wachspress rational functions is that they are linear on the finite element boundary between two neighboring corner nodes. Additionally, orthogonal stress and strain functions are used in the mixed formulation. This eliminates the problem of time consuming numerical inversions of matrices at the element level. The performance of the enhanced elements will be illustrated in a series of test examples.

Proceedings of the 11th EAI International Conference on Mobile Multimedia Communications, 2018

In many disciplines there is a need for efficient in terpolation of ir regular spaced data. For u... more In many disciplines there is a need for efficient in terpolation of ir regular spaced data. For unsampled locations values have to be computed from the available data. Usually we are interested in smooth interpolations, and artificial o scillations should be avoided. For large data sets fully populated matrices are undesirable. The use of compactly supported basis functions appears to be attractive. In this paper radial basis functions satisfying the 2D bi-harmonic equation are used. Accurate interpolation could be achieved in the numerical test examples.

Thirteenth International Symposium on Temporal Representation and Reasoning (TIME'06), 2006

ABSTRACT This paper considers a set of ozone data in the conterminous U.S., which records the ozo... more ABSTRACT This paper considers a set of ozone data in the conterminous U.S., which records the ozone concentration levels at a set of monitoring sites during 1994 and 1999. Existing GIS techniques are insufficient in handling such kind of spatiotemporal data in terms of data interpolation, visualization, representation and querying. We adopt 3-D shape functions from finite element methods for the spatiotemporal interpolation of the ozone dataset and analyze interpolation errors. The 3-D shape function based method estimates ozone concentration levels with less than 10 percent Mean Absolute Percentage Error. We give two approaches for visualizing the data: (i) combining the ArcGIS visualization tool with shape function interpolation results to visualize the ozone data for each year from 1994 and 1999, (ii) using Matlab to visualize the interpolated ozone data in a 3-D vertical profile display. For the spatiotemporal data representation, we use the constraint data model, because it can give an efficient and accurate representation of interpolation results. Finally, we give some practical query examples.

ABSTRACT For stress concentrations around voids and inclusions, one can derive appropriate stress... more ABSTRACT For stress concentrations around voids and inclusions, one can derive appropriate stress and strain functions, which characterize the local fields very well, and use them for special problem adapted fi-nite elements. For linear problems, such stress and strain functions can be obtained from complex solution representations. The functions can be derived in such a way that the equilibrium equations and the boundary conditions on the void surface or the continuity conditions along the matrix/inclusion boundary are satisfied a priori. For special finite elements with voids and inclusions, piecewise lin-ear or quadratic boundary displacements have to be assumed for an appropriate coupling with other finite elements. There has to be a balance between the number of stress/strain parameters and the nodal displacements of the special elements with built-in voids or inclusions. The outer boundary of a two-dimensional special finite element can be a polygon, and for the three-dimensional case the element boundary can be chosen as a polyhedron. INTRODUCTION Stress concentrations occur in the neighborhood of holes and inclusions. For an accurate prediction of the stress peaks, a fine mesh of standard displacement finite elements is required for an accurate prediction of the stress peaks. For the modeling of heterogeneous materials with numerous inclusions a very large number of standard finite elements can be required in a simu-lation. In order to reduce the number of elements necessary in a numerical simulation, the use of special problem adapted finite elements with built-in inclusions or voids (Fig. 1) is attractive (Fig. 2). For such elements trial functions for stresses and strains can be constructed which satisfy the governing differential equations and boundary conditions on the void or inclusion surface a priori. Since functions are used, which satisfy the governing differential equations, the procedure can be seen as a Trefftz method (Trefftz, 1926). The Trefftz method has been utilized for the derivation of Trefftz-type finite elements by several researchers, as for exam-ple,. In 1982 Piltner proposed special finite elements with elliptical holes, internal and external cracks. For the modeling of heterogeneous materials Zhang, Katsube and Zeng, Ghosh and Moorthy utilized special elements with built-in inclusions. Ghosh and his co-workers developed finite elements with element boundaries generated during a Voronoi tessalation process. In (Fig. 2) a discretization with special Voronoi elements is shown.

This conference proceeding was published in Proceedings of the US National Congress on Computatio... more This conference proceeding was published in Proceedings of the US National Congress on Computational Mechanics

This paper investigates spatiotemporal interpolation methods for the application of air pollution... more This paper investigates spatiotemporal interpolation methods for the application of air pollution assessment. The air pollutant of interest in this paper is fine particulate matter PM2.5. The choice of the time scale is investigated when applying the shape function-based method. It is found that the measurement scale of the time dimension has an impact on the interpolation results. Based upon the comparison between the accuracies of interpolation results, the most effective time scale out of four experimental ones was selected for performing the PM2.5 interpolation. The paper also evaluates the population exposure to the ambient air pollution of PM2.5 at the county-level in the contiguous U.S. in 2009. The interpolated county-level PM2.5 has been linked to 2009 population data and the population with a risky PM2.5 exposure has been estimated. The risky PM2.5 exposure means the PM2.5 concentration exceeding the National Ambient Air Quality Standards. The geographic distribution of the counties with a risky PM2.5 exposure is visualized. This work is essential to understanding the associations between ambient air pollution exposure and population health outcomes.

Advances in Engineering Software, 1995

In 1926 E. Trefftz published a paper about a variational formulation which utilizes boundary inte... more In 1926 E. Trefftz published a paper about a variational formulation which utilizes boundary integrals. Almost half a century later researchers became interested again in the ideas of Trefftz when the potential advantage of the Trefftz-method for an efficient use in numerical ...

Solution representations are available for severeal differential equations. For elasticity proble... more Solution representations are available for severeal differential equations. For elasticity problems some of the solution representations are considered in this paper. The solution representations can be used for a systematic construction of Trefftz functions for the derivation of Trefftz-type finite elements. For the example of a thick plate a set of Trefftz functions is presented.

Engineering Analysis with Boundary Elements

For the Trefftz method sequences of linear independent functions satisfying the governing differe... more For the Trefftz method sequences of linear independent functions satisfying the governing differential equations are needed. For two- and three-dimensional elasticity problems some useful options for obtaining Trefftz trial functions are discussed. The discussion also includes some very useful particular solutions. For three-dimensional elasticity problems four methods are considered: the displacement representation of Papkovich/Neuber, Piltner's complex representation, a hypercomplex displacement representation of Bock, G{\"u}rlebeck, Weisz-Patrault, Legatiuk, and H.M. Nguyen, as well as Slobodyanskii's representation written in real and complex form. For two-dimensional problems the option of using discretized Cauchy integrals is illustrated. Very briefly Trefftz type Radial Basis Functions are mentioned satisfying a homogeneous or inhomogeneous differential equation. This paper is not meant as a complete survey or review on Trefftz methods. The paper presents a collection of personal choices to help future developers of numerical methods based on Trefftz trial functions. Keywords: Trefftz method, Radial Basis Functions, Discretized Cauchy integrals, Elasticity solutions, Data interpolation.

Boundary Integral Methods, 1991

Boundary Element Methods in Engineering, 1990

ABSTRACT

Mechanics Research Communications, 1991

ABSTRACT The analysis of a plate as a three-dimensional continuum leads to the representation of ... more ABSTRACT The analysis of a plate as a three-dimensional continuum leads to the representation of the displacements and stresses in terms of functions which must satisfy characteristic differential equations. Membrane and bending cases are considered separately. The application of the representations allows the inclusion of warping for plate bending and leads to very accurate results in numerical computations.

International Journal of Environmental Research and Public Health, 2016

Appropriate spatiotemporal interpolation is critical to the assessment of relationships between e... more Appropriate spatiotemporal interpolation is critical to the assessment of relationships between environmental exposures and health outcomes. A powerful assessment of human exposure to environmental agents would incorporate spatial and temporal dimensions simultaneously. This paper compares shape function (SF)-based and inverse distance weighting (IDW)-based spatiotemporal interpolation methods on a data set of PM 2.5 data in the contiguous U.S. Particle pollution, also known as particulate matter (PM), is composed of microscopic solids or liquid droplets that are so small that they can get deep into the lungs and cause serious health problems. PM 2.5 refers to particles with a mean aerodynamic diameter less than or equal to 2.5 micrometers. Based on the error statistics results of k-fold cross validation, the SF-based method performed better overall than the IDW-based method. The interpolation results generated by the SF-based method are combined with population data to estimate the population exposure to PM 2.5 in the contiguous U.S. We investigated the seasonal variations, identified areas where annual and daily PM 2.5 were above the standards, and calculated the population size in these areas. Finally, a web application is developed to interpolate and visualize in real time the spatiotemporal variation of ambient air pollution across the contiguous U.S. using air pollution data from the U.S. Environmental Protection Agency (EPA)'s AirNow program.