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Published in Ansel C. Ugural, Youngjin Chung, Errol A. Ugural, Mechanical Engineering Design, 2020
Ansel C. Ugural, Youngjin Chung, Errol A. Ugural
We now develop the global stiffness matrix for an element oriented arbitrarily in a 2D plane. The local coordinates are chosen to conveniently represent the individual element, whereas the global or reference coordinates are chosen to be convenient for the whole structure. We designate the local and global coordinate systems for an axial element by x¯,y¯ and x, y, respectively (Figure 17.4).
A GIS-based simulation and visualization tool for the assessment of gully erosion processes
Published in Journal of Spatial Science, 2022
Adel Omran, Dietrich Schröder, Christian Sommer, Volker Hochschild, Michael Märker
The last step regards the visualisation of the model output in a GIS environment. The aim of this stage is to embed the calculated gully depths and widths values along the flow line into a coordinate system in order to be visualised in a 2D and/or a 3D visualisation format, together with the related topography. This embedding is achieved by calculating 3D polygon features representing the gully bottom and the gully walls between adjacent flow line points for each time step, e.g. for each year. These polygons were created based on the defined coordinate file of the study area calculated in the data preparation step, and the geometrical parameters that were calculated in the simulation. Using simple geometric calculations implemented in a Python script, the local coordinates along the flow line for each layer are transformed into coordinates (Easting and Northing values) related to a geographic coordinate system. Additionally, the script creates 3D polygon geometries based on the GIS API for each year. The polygons can be projected and merged to define the upper boundary of gully erosion. As an output of this script, we achieve sequences of polygons that represent the bottoms and side walls of the stream flow lines for each layer after the erosion process at each time steps (e.g. yearly). The creation of these sequences of polygon geometries is used in the further 2D and 3D output visualisation of gully erosion in the GIS environment.
Control volume finite element method for entropy generation minimization in mixed convection of nanofluids
Published in Numerical Heat Transfer, Part B: Fundamentals, 2019
The governing equations for this problem are discretized with a control-volume-based finite element method (CVFEM) [42]. The mesh is discretized based on the finite element method. Then, the subcontrol volumes are assembled from the neighboring adjacent elements to form a control volume around the nodal points. Consider a local coordinate system which describes the shape functions and properties of the element as shown in Figure 3. The problem domain is discretized into linear quadrilateral finite elements. The control volume is defined by the subsurfaces and subcontrol volumes of the neighboring four elements. The conservation equations are then integrated over the finite control volumes and time steps to obtain the discretized conservation equations. The control volumes are further subdivided into four subcontrol volumes within each element, which are linked with a control volume and its respective element node. The subsurfaces are the boundaries of the control volumes, which coincide with the exterior boundaries of the element. The local coordinates are defined by a local coordinate system The midpoint of a subsurface is defined by the integration point
REDIM reduced modeling of flame quenching at a cold wall – The influence of detailed transport models and detailed mechanisms
Published in Combustion Science and Technology, 2019
Christina Strassacker, Viatcheslav Bykov, Ulrich Maas
Contrary to the computation of a reacting system with detailed kinetics, where the evolution equation is given by partial differential equations for the state vector , the state vector for the computation with the REDIM method can be reduced to dimensions (). It is defined as , with the parametrization vector , which represents the local coordinates on the manifold. Here, represents the specific enthalpy, the pressure, and the specific mole fraction consisting of mass fraction and the molar mass of the species .