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Erschienen in:
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2014 | OriginalPaper | Buchkapitel

15. Specific Topics

verfasst von : Hans-Jörg G. Diersch

Erschienen in: FEFLOW

Verlag: Springer Berlin Heidelberg

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Abstract

The finite element solution of the governing flow, mass and heat transport equations as described in the preceding chapters requires the discretization of the equations to replace the continuous PDE’s with a system of simultaneous algebraic equations (cf. Chap.​ 8). The spatial discretization is accomplished by subdividing the study domain with its boundary into a number of nonoverlapping finite elements of different shapes, such as triangles, tetrahedra, bricks (see Fig. 8.6), forming the finite element mesh associated with a set of nodes and interpolation functions.

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Vorheriges Kapitel Discrete Feature Modeling of Flow, Mass and Heat Transport Processes
Fußnoten
1
Considering a superelement side of length L and move the midside node to the distance aL, where \(0 \leq a \leq \tfrac{1} {2}\) is a shifting factor, viz.,
https://static-content.springer.com/image/chp%3A10.1007%2F978-3-642-38739-5_15/MediaObjects/209325_1_En_15_Un1_HTML.gif
the smallest element length Δ x of the graded element spacing obtained with the parabolic mapping is:
$$\displaystyle{\varDelta x =\varDelta \xi L[\varDelta \xi (\tfrac{1} {2} - a) + 2a -\tfrac{1} {2}]}$$
where Δ ξ > 0 is the given increment (15.3) of the superelement side subdivision. It results by taking the parabolic interpolation functions of Tab. G.1(b) of Appendix G with \(\xi = -1+\varDelta \xi\) for the second evaluation point. For \(a = \tfrac{1} {2}\) the standard equally graded spacing with \(\varDelta x = \tfrac{1} {2}\varDelta \xi L\) is given, while with the a midside node shift of \(a = \tfrac{1} {4}\) a left-sided densification with \(\varDelta x ={ \tfrac{1} {4}\varDelta \xi }^{2}L\) results. Since \(\varDelta \xi (\tfrac{1} {2} - a) + 2a -\tfrac{1} {2}\) must be positive, the following constraints are required:
$$\displaystyle{\begin{array}{rclll} a& >&\frac{1} {2}\Bigl (\frac{1-\varDelta \xi } {2-\varDelta \xi }\Bigr ) &\quad \mbox{ for} & \quad 0 \leq \varDelta \xi \leq 1 \\ \varDelta \xi & >&\frac{1-4a} {1-2a} &\quad \mbox{ for} & \quad 0 \leq a \leq \frac{1} {4} \end{array} }$$
We recognize that with decreasing Δ ξ → 0 the shift of the midside node must satisfy \(a > \tfrac{1} {4}\).
 
2
Boundary nodes \(\boldsymbol{x}_{i}\) (i = 1, 2, …) are created on each side of a superelement. Their distances depend on the desired element resolution. They can be equally distributed along the superelement side or can be densified locally by using a parabolic grading function:
$$\displaystyle{\boldsymbol{x}_{i} =\boldsymbol{ x}_{2} +\xi _{i}(k)[\boldsymbol{a} +\xi _{i}(k)\boldsymbol{b}],\quad (-5 \leq k \leq 5)}$$
with
$$\displaystyle{\boldsymbol{a} = \tfrac{1} {2}(\boldsymbol{x}_{3} -\boldsymbol{ x}_{1}),\quad \boldsymbol{b} = \tfrac{1} {2}(\boldsymbol{x}_{3} +\boldsymbol{ x}_{1}) -\boldsymbol{ x}_{2}}$$
$$\displaystyle{\xi _{i}(k) =\eta _{i} - s(k)(\eta _{i}^{2} - 1),\quad \eta _{ i} = -1 + (i - 1)\varDelta \eta,\quad s(k) = \tfrac{1} {4}\mathrm{sgn}(k)\sum _{j=1}^{\vert k\vert }{2}^{-j+1}}$$
where \(\boldsymbol{x}_{1}\), \(\boldsymbol{x}_{2}\) and \(\boldsymbol{x}_{3}\) are the coordinates of the left, middle and right nodes of a superelement side, respectively, k is a grading counter (for k = 0 there is no grading and the nodes become equally distributed, k > 0 leads to left-sided densification, k < 0 leads to a right-sided densification of boundary nodes) and \(\varDelta \eta = 2/(\mathrm{NS} + 1)\) is a local coordinate increment determined by the desired number of superelement side segmentation NS.
 
3
Alternatively, to find an appropriate energy expression for coupled variable-density flow, mass and heat transport in porous media, the internal Clausius-Duhem entropy production ρ Υ ≥ 0, (3.​125), can be utilized. It provides a physically consistent functional in which all relevant state variables of the nonlinearly coupled process in form of Darcy flux \(\boldsymbol{q}\), hydraulic head h, species mass C k and temperature T are present. A simplified version of (3.​125) yields
$$\displaystyle{\bar{\varUpsilon }(\boldsymbol{q},T,C_{k}) = T\,\rho \varUpsilon =\rho _{0}g\boldsymbol{q} \cdot (\boldsymbol{{K}}^{-1} \cdot \boldsymbol{ q}) + \frac{1} {T}\bigl (\nabla T \cdot (\boldsymbol{\varLambda }\cdot \nabla T)\bigr ) +\sum _{k} \frac{\partial \mu _{k}} {\partial C_{k}}\bigl (\nabla C_{k} \cdot (\boldsymbol{D}_{k} \cdot \nabla C_{k})\bigr ) \geq 0}$$
and the following entropy error norm appears suitable
$$\displaystyle{\Vert \boldsymbol{{e}\Vert }^{2} =\int _{\varOmega }\bar{\varUpsilon }(\boldsymbol{q} -\hat{\boldsymbol{ q}},T -\hat{ T},C_{ k} -\hat{ C}_{k})d\varOmega }$$
where \(\boldsymbol{q}\), C k , T are the exact solutions and \(\hat{\boldsymbol{q}}\), \(\hat{C}_{k}\), \(\hat{T}\) are the approximate finite element solutions. The Darcy flux \(\boldsymbol{q} =\boldsymbol{ q}(h,C_{k},T)\) or \(\hat{\boldsymbol{q}} =\hat{\boldsymbol{ q}}(\hat{h},\hat{C}_{k},\hat{T})\) takes the form of (11.​1).
 
4
Equivalently, for 2D finite elements the Newton iteration scheme results
$$\displaystyle{\begin{array}{rclcl} F_{1}{(\boldsymbol{\eta }}^{\tau +1}) & = & F_{1}{(\boldsymbol{\eta }}^{\tau }) + \frac{\partial F_{1}{(\boldsymbol{\eta }}^{\tau })} {{\partial \xi }^{\tau }} {(\xi }^{\tau +1} {-\xi }^{\tau }) + \frac{\partial F_{1}{(\boldsymbol{\eta }}^{\tau })} {{\partial \eta }^{\tau }} {(\eta }^{\tau +1} {-\eta }^{\tau }) - {x}^{e}& = & 0 \\ F_{2}{(\boldsymbol{\eta }}^{\tau +1}) & = & F_{2}{(\boldsymbol{\eta }}^{\tau }) + \frac{\partial F_{2}{(\boldsymbol{\eta }}^{\tau })} {{\partial \xi }^{\tau }} {(\xi }^{\tau +1} {-\xi }^{\tau }) + \frac{\partial F_{2}{(\boldsymbol{\eta }}^{\tau })} {{\partial \eta }^{\tau }} {(\eta }^{\tau +1} {-\eta }^{\tau }) - {y}^{e} & = & 0 \end{array} }$$
or
$$\displaystyle{\left (\begin{array}{cccc} a_{11} & a_{12}\\ a_{21 } & a_{22} \end{array} \right )\cdot \left (\begin{array}{cccc} { \varDelta \xi }^{\tau }\\ { \varDelta \eta }^{\tau } \end{array} \right ) = \left (\begin{array}{cccc} a_{13} \\ a_{23} \end{array} \right )}$$
and
$$\displaystyle{\begin{array}{rcl} { \varDelta \xi }^{\tau } & = & \frac{1} {\vert a\vert }\bigl (a_{13}a_{22} - a_{23}a_{12}\bigr ) \\ { \varDelta \eta }^{\tau } & = & \frac{1} {\vert a\vert }\bigl (a_{11}a_{23} - a_{21}a_{13}\bigr ) \\ \mbox{ with} & & \\ \vert a\vert & = &a_{ 11}a_{22} - a_{21}a_{12}\neq 0 \end{array} }$$
 
Literatur
16.
Babuška, I.: Reliability of computational mechanics. In: Whiteman, J. (ed.) The Mathematics of Finite Elements and Applications: Highlights 1993, pp. 25–44. Wiley, Chichester (1994)
23.
24.
Bank, R., Sherman, A., Weiser, A.: Refinement algorithms and data structure for regular local mesh refinement. In: Steplemen, R., et al. (eds.) Scientific Computing, pp. 3–17. IMACS/North Holland, Brussels (1983)
50.
Bowyer, A.: Computing Dirichlet tesselations. Comput. J. 24(2), 162–166 (1981)CrossRef
64.
Burkhart, D., Hamann, B., Umlauf, G.: Adaptive and feature-preserving subdivision for high-quality tetrahedral meshes. Comput. Graph. Forum 29(1), 117–127 (2010)CrossRef
99.
Cordes, C., Kinzelbach, W.: Continuous groundwater velocity fields and path lines in linear, bilinear, and trilinear finite elements. Water Resour. Res. 28(11), 2903–2911 (1992)CrossRef
154.
Elder, J.: Transient convection in a porous medium. J. Fluid Mech. 27(3), 609–623 (1967)CrossRef
155.
Engel, B., Navulur, K.: The role of geographical information systems in groundwater modeling (Chapter 21). In: Delleur, J. (ed.) The Handbook of Groundwater Engineering, pp. 21:1–16. CRC/Springer, Boca Raton (1999)
165.
Fletcher, C.: Computational Techniques for Fluid Dynamics, vols. 1 and 2. Springer, New York (1988)
169.
Franca, A., Haghighi, K.: Adaptive finite element analysis of transient thermal problems. Numer. Heat Transf. Part B 26(3), 273–292 (1994)CrossRef
188.
George, P.: Automatic Mesh Generation: Application to Finite Element Methods. Wiley, Chichester (1991)
222.
Hægland, H., Dahle, H., Eigestad, G., Lie, K.A., Aavatsmark, I.: Improved streamlines and time-of-flight for streamline simulation on irregular grids. Adv. Water Resour. 30(4), 1027–1045 (2007)CrossRef
263.
Houlding, S.: 3D Geoscience Modeling. Springer, Berlin/Heidelberg (1994)CrossRef
294.
Jourde, H., Cornaton, F., Pistre, S., Bidaux, P.: Flow behavior in a dual fracture network. J. Hydrol. 266(1–2), 99–119 (2002)CrossRef
314.
Knupp, P., Steinberg, S.: Fundamentals of Grid Generation. CRC, Boca Raton (1994)
347.
Lian, Y.Y., Hsu, K.H., Shao, Y.L., Lee, Y.M., Jeng, Y.W., Wu, J.S.: Parallel adaptive mesh-refining scheme on a three-dimensional unstructured tetrahedral mesh and its applications. Comput. Phys. Commun. 175(11–12), 721–737 (2006)CrossRef
353.
Löhner, R.: Applied CFD Techniques. Wiley, Chichester (2001)
369.
372.
Merchant, M., Weatherill, N.: Adaptivity techniques for compressible inviscid flows. Comput. Methods Appl. Mech. Eng. 106(1–2), 83–106 (1993)CrossRef
402.
Oñate, E., Bugeda, G.: Mesh optimality criteria for adaptive finite element computations. In: Whiteman, J. (ed.) The Mathematics of Finite Elements and Applications – Highlights 1993, pp. 121–135. Wiley, Chichester (1994)
419.
Pinder, G.: Groundwater Modeling Using Geographical Information Systems. Wiley, New York (2002)
424.
Pokrajac, D., Lazic, R.: An efficient algorithm for high accuracy particle tracking in finite elements. Adv. Water Resour. 25(4), 353–369 (2002)CrossRef
425.
Pollock, D.: Semianalytical computation of path lines for finite-difference models. Groundwater 26(6), 743–750 (1988)CrossRef
430.
Press, W., Flannery, B., Teukolsky, S., Vetterling, W.: Numerical Recipes in C. Cambridge University Press, Cambridge (1988)
433.
Rannacher, R., Bangerth, W.: Adaptive Finite Element Methods for Differential Equations. Birkhäuser, Basel (2003)
434.
Raper, J.: Three Dimensional Applications in Geographic Information Systems. Taylor and Francis, London (1989)
455.
Sadek, E.: A scheme for the automatic generation of triangular finite elements. Int. J. Numer. Methods Eng. 15(12), 1813–1822 (1980)CrossRef
475.
501.
Suárez, J., Abad, P., Plaza, A., Padrón, M.: Computational aspects of the refinement of 3D tetrahedral meshes. J. Comput. Methods Sci. Eng. 5(4), 215–224 (2005)
514.
Thompson, J., Soni, B., Weatherill, N.: Handbook of Grid Generation. CRC, Boca Raton (1999)
545.
Verfürth, R.: A Review of a Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques. Wiley/Teubner, New York/Stuttgart (1996)
557.
Watson, D.: Computing n-dimensional Delaunay tesselation with application to Voronoi polytopes. Comput. J. 24(2), 167–172 (1981)CrossRef
587.
Zheng, C., Bennett, G.: Applied Contaminant Transport Modeling: Theory and Practice. Van Nostrand Reinhold, New York (1995)
590.
Zienkiewicz, O., Taylor, R.: The Finite Element Method. Volume 1: The Basis, 5th edn. Butterworth-Heinemann, Oxford (2000)
593.
Zienkiewicz, O., Zhu, J.: A simple error estimator and adaptive procedure for practical engineering analysis. Int. J. Numer. Methods Eng. 24(2), 337–357 (1987)CrossRef
594.
Zienkiewicz, O., Zhu, J.: The superconvergent patch recovery and a posteriori error estimates. Part 1: The recovery technique. Int. J. Numer. Methods Eng. 33(7), 1331–1364 (1992)
Metadaten
Titel
Specific Topics
verfasst von
Hans-Jörg G. Diersch
Copyright-Jahr
2014
Verlag
Springer Berlin Heidelberg
Buch
FEFLOW

Print ISBN: 978-3-642-38738-8

Electronic ISBN: 978-3-642-38739-5

Copyright-Jahr: 2014

DOI
https://doi.org/10.1007/978-3-642-38739-5_15