Spatial Behavior for Nonlinear Heat Equations

1997 ◽  
Vol 07 (05) ◽  
pp. 633-647 ◽  
Author(s):  
R. Quintanilla
Keyword(s):  
Cross Sections ◽  
Order Approximation ◽  
Three Dimensional ◽  
Spatial Behavior ◽  
Nonlinear Heat Equation ◽  
Heat Equations ◽  
Decay Estimates ◽  
Spatial Estimates ◽  
Comparison Arguments ◽  
First Order Method

In this paper we investigate spatial decay and growth estimates for the solutions of the nonlinear heat equation in a three-dimensional cylinder with homogeneous Dirichlet conditions prescribed on the lateral surface for all time. We derive two Phragmen–Lindelöf type growth–decay estimates. Two methods are used in our approximation. One of them involves a measure on the cross-sections and the other uses the "energy" contained in the part of the cylinder beyond a cross-section. For suitable volumetric heat capacity and thermal conductivitythe second-order approximation combined with comparison arguments allow us to improve the decay estimates. We also sketch the application of the first-order method in the case where the solid is a cone. Spatial estimates for the backward nonlinear equation are presented in the last section.

REFINED BEAM THEORIES BASED ON A UNIFIED FORMULATION

2010 ◽  
Vol 02 (01) ◽  
pp. 117-143 ◽  
Author(s):  
ERASMO CARRERA ◽  
GAETANO GIUNTA
Keyword(s):  
Differential Equations ◽  
Cross Section ◽  
Cross Sections ◽  
Order Approximation ◽  
Three Dimensional ◽  
Closed Form Solution ◽  
Approximation Order ◽  
Form Solution ◽  
Geometrical Parameters ◽  
Beam Theories

This paper proposes several axiomatic refined theories for the linear static analysis of beams made of isotropic materials. A hierarchical scheme is obtained by extending plates and shells Carrera's Unified Formulation (CUF) to beam structures. An N-order approximation via Mac Laurin's polynomials is assumed on the cross-section for the displacement unknown variables. N is a free parameter of the formulation. Classical beam theories, such as Euler-Bernoulli's and Timoshenko's, are obtained as particular cases. According to CUF, the governing differential equations and the boundary conditions are derived in terms of a fundamental nucleo that does not depend upon the approximation order. The governing differential equations are solved via the Navier type, closed form solution. Rectangular and I-shaped cross-sections are accounted for. Beams undergo bending and torsional loadings. Several values of the span-to-height ratio are considered. Slender as well as deep beams are analysed. Comparisons with reference solutions and three-dimensional FEM models are given. The numerical investigation has shown that the proposed unified formulation yields the complete three-dimensional displacement and stress fields for each cross-section as long as the appropriate approximation order is considered. The accuracy of the solution depends upon the geometrical parameters of the beam and loading conditions.

scholarly journals A 3-D Pseudo-Rigid Body Model for Rectangular Cantilever Beams With an Arbitrary Force End-Load

2014 ◽  
Author(s):  
Jairo Chimento ◽  
Craig Lusk ◽  
Ahmad Alqasimi
Keyword(s):  
Rigid Body ◽  
Cross Sections ◽  
Degrees Of Freedom ◽  
Three Dimensional ◽  
Neutral Axis ◽  
Spatial Behavior ◽  
Model Parameters ◽  
Cantilever Beams ◽  
Body Model ◽  
Rigid Body Model

This paper presents the first three-dimensional pseudo-rigid body model (3-D PRBM) for straight cantilever beams with rectangular cross sections and spatial motion. Numerical integration of a system of differential equations yields approximate displacement and orientation of the beam’s neutral axis at the free-end, and curvatures of the neutral axis at the fixed-end. This data was used to develop the 3-D PRBM which consists of two torsional springs connecting two rigid links for a total of 2 degrees of freedom (DOF). The 3-D PRBM parameters that are comparable with existing 2-D model parameters are characteristic radius factor (means: γ = 0.8322), bending stiffness coefficient (means: KΘ = 2.5167) and parametric angle coefficient (means: cΘ = 1.2501). New parameters are introduced in the model in order to capture the spatial behavior of the deflected beam including two parametric angle coefficients (means: cΨ = 1.0714; cΦ = 1.0087).

Geological Field Sketches and Illustrations

2019 ◽  
Author(s):  
Matthew J. Genge
Keyword(s):  
Cross Sections ◽  
Earth Science ◽  
Three Dimensional ◽  
Worked Examples ◽  
Scientific Publications ◽  
Thin Sections ◽  
Geological Features ◽  
Different Types ◽  
The Subject

Drawings, illustrations, and field sketches play an important role in Earth Science since they are used to record field observations, develop interpretations, and communicate results in reports and scientific publications. Drawing geology in the field furthermore facilitates observation and maximizes the value of fieldwork. Every geologist, whether a student, academic, professional, or amateur enthusiast, will benefit from the ability to draw geological features accurately. This book describes how and what to draw in geology. Essential drawing techniques, together with practical advice in creating high quality diagrams, are described the opening chapters. How to draw different types of geology, including faults, folds, metamorphic rocks, sedimentary rocks, igneous rocks, and fossils, are the subjects of separate chapters, and include descriptions of what are the important features to draw and describe. Different types of sketch, such as drawings of three-dimensional outcrops, landscapes, thin-sections, and hand-specimens of rocks, crystals, and minerals, are discussed. The methods used to create technical diagrams such as geological maps and cross-sections are also covered. Finally, modern techniques in the acquisition and recording of field data, including photogrammetry and aerial surveys, and digital methods of illustration, are the subject of the final chapter of the book. Throughout, worked examples of field sketches and illustrations are provided as well as descriptions of the common mistakes to be avoided.

scholarly journals Thermo-Hydraulic Performance of U-Tube Borehole Heat Exchanger with Different Cross-Sections

2021 ◽  
Vol 13 (6) ◽  
pp. 3255
Author(s):  
Aizhao Zhou ◽  
Xianwen Huang ◽  
Wei Wang ◽  
Pengming Jiang ◽  
Xinwei Li
Keyword(s):  
Heat Transfer ◽  
Heat Exchange ◽  
Heat Resistance ◽  
Cross Sections ◽  
Three Dimensional ◽  
Thermal Response ◽  
Transfer Efficiency ◽  
Cross Sectional ◽  
Heat Transfer Efficiency ◽  
Oval Tube

For reducing the initial GSHP investment, the heat transfer efficiency of the borehole heat exchange (BHE) system can be enhanced to reduce the number or depth of drilling. This paper proposes a novel and simple BHE design by changing the cross-sectional shape of the U-tube to increase the heat transfer efficiency of BHEs. Specifically, in this study, we (1) verified the reliability of the three-dimensional numerical model based on the thermal response test (TRT) and (2) compared the inlet and outlet temperatures of the different U-tubes at 48 h under the premise of constant leg distance and fluid area. Referent to the circular tube, the increases in the heat exchange efficiencies of the curved oval tube, flat oval tube, semicircle tube, and sector tube were 13.0%, 19.1%, 9.4%, and 14.8%, respectively. (3) The heat flux heterogeneity of the tubes on the inlet and outlet sides of the BHE, in decreasing order, is flat oval, semicircle, curved oval, sector, and circle shapes. (4) The temperature heterogeneity of the borehole wall in the BHE in decreasing order is circle, sector, curved oval, flat oval, and semicircle shapes. (5) Under the premise of maximum leg distance, referent to the heat resistance of the tube with a circle shape at 48 h, the heat exchange efficiency of the curved oval, flat oval, semicircle, and sector tubes increased 12.6%, 17.7%, 10.3%, and 7.8%, respectively. (6) We found that the adjustments of the leg distance and the tube shape affect the heat resistance by about 25% and 12%, respectively. (7) The flat-oval-shaped tube at the maximum leg distance was found to be the best tube design for BHEs.

scholarly journals Computer-aided design of multi-purpose steel headframes for mines with a new technical level

2020 ◽  
Vol 174 ◽  
pp. 01048
Author(s):  
Elena Kassikhina ◽  
Vladimir Pershin ◽  
Nina Rusakova
Keyword(s):  
Cross Sections ◽  
Computer Aided Design ◽  
Information Technologies ◽  
Three Dimensional ◽  
Numerical Control ◽  
Technical Level ◽  
Resource Saving ◽  
Existing Structures ◽  
Set Up ◽  
Aided Design

The existing structures of the steel sinking headgear and permanent headframe do not meet the requirements of resource saving (metal consumption and manpower input at installation), and the present methods of the headframe designing do not fully reflect recent possibilities of applying of the advanced information technologies. Technical level of the modern software makes it possible for designers to set up multiple numerical experiments to create a computer simulation that allows solving the problem without field and laboratory experiments, and therefore without special costs. In this regard, a mathematical simulation has been developed and based on it, software to select cross-sections of multi- purpose steel headframe elements and to calculate proper weight of its metal structures depending on the characteristics and hoisting equipment. A headframe drawing is displayed, as the results of the software work, including list of elements, obtained optimal hoisting equipment in accordance with the initial data. The software allows speeding up graphic work and reducing manpower input on calculations and paper work. The software allows developing a three-dimensional image of the structure and its functional blocks, based on the obtained initial parameters, as well as developing control software for units with numerical control (NC) in order to manufacture multi-purpose headframes.

Analysing the Natural Resting Aspects of a Component for Automated Assembly

1995 ◽  
Vol 209 (2) ◽  
pp. 125-131 ◽  
Author(s):  
B K A Ngoi ◽  
L E N Lim ◽  
S S G Lee ◽  
S W Lye
Keyword(s):  
Cross Sections ◽  
Computer Aided Design ◽  
Energy Barrier ◽  
Three Dimensional ◽  
Rectangular Prism ◽  
Automatic Assembly ◽  
Automated Assembly ◽  
Barrier Method ◽  
Aided Design ◽  
Natural Resting Aspect

This paper proposes the construction of an energy envelope that can be used to advantage with the energy barrier method to analyse the natural resting aspect of engineering parts destined for automatic assembly. Unlike the energy barrier method, the energy envelope does not require any visualization of the projection of the energy barrier on the aspect of interest. The energy envelope is the three-dimensional topology of the changes in height of the centroid, as the part attempts changes of aspect. The paper describes how it may be computed in a CAD (computer aided design) solid modeller. The results of applying the energy envelope to prisms of square and cylindrical cross-sections are the same as those predicted by the energy barrier method. When extended to the analysis of a rectangular prism, the results were consistent with Boothroyd's dynamic solution and Boothroyd's experimental data. This conclusion is encouraging as there is no irrefutable evidence that the energy barrier method may be applied to the analysis of the rectangular prism.

Sign in / Sign up

Export Citation Format

Share Document