Skip to main content
SpringerLink
Log in

Specific Topics

  • Chapter
  • First Online:
FEFLOW

  • 4608 Accesses

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.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 229.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 299.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 299.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 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.,

    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. 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. 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. 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

    $$\displaystyle{\begin{array}{rcl} { \varDelta \xi }^{\tau } & = & \frac{1} {\vert a\vert }\bigl (a_{13}a_{22} - a_{23}a_{12}\bigr ) \\ { \varDelta \eta }^{\tau } & = & \frac{1} {\vert a\vert }\bigl (a_{11}a_{23} - a_{21}a_{13}\bigr ) \\ \mbox{ with} & & \\ \vert a\vert & = &a_{ 11}a_{22} - a_{21}a_{12}\neq 0 \end{array} }$$

References

  1. 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)

    Google Scholar 

  2. Bank, R.: PLTMG: a software package for solving elliptic partial differential equations – user’s guide 11.0. Technical report, Department of Mathematics, University of California at San Diego, La Jolla (2012). http://ccom.ucsd.edu/~reb/software.html

  3. 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)

    Google Scholar 

  4. Bowyer, A.: Computing Dirichlet tesselations. Comput. J. 24(2), 162–166 (1981)

    Article  Google Scholar 

  5. Burkhart, D., Hamann, B., Umlauf, G.: Adaptive and feature-preserving subdivision for high-quality tetrahedral meshes. Comput. Graph. Forum 29(1), 117–127 (2010)

    Article  Google Scholar 

  6. 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)

    Article  Google Scholar 

  7. Elder, J.: Transient convection in a porous medium. J. Fluid Mech. 27(3), 609–623 (1967)

    Article  Google Scholar 

  8. 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)

    Google Scholar 

  9. Fletcher, C.: Computational Techniques for Fluid Dynamics, vols. 1 and 2. Springer, New York (1988)

    Google Scholar 

  10. Franca, A., Haghighi, K.: Adaptive finite element analysis of transient thermal problems. Numer. Heat Transf. Part B 26(3), 273–292 (1994)

    Article  Google Scholar 

  11. George, P.: Automatic Mesh Generation: Application to Finite Element Methods. Wiley, Chichester (1991)

    Google Scholar 

  12. 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)

    Article  Google Scholar 

  13. Houlding, S.: 3D Geoscience Modeling. Springer, Berlin/Heidelberg (1994)

    Book  Google Scholar 

  14. Jourde, H., Cornaton, F., Pistre, S., Bidaux, P.: Flow behavior in a dual fracture network. J. Hydrol. 266(1–2), 99–119 (2002)

    Article  Google Scholar 

  15. Knupp, P., Steinberg, S.: Fundamentals of Grid Generation. CRC, Boca Raton (1994)

    Google Scholar 

  16. 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)

    Article  Google Scholar 

  17. Löhner, R.: Applied CFD Techniques. Wiley, Chichester (2001)

    Google Scholar 

  18. McLaren, R.: GRIDBUILDER: a generator for 2D triangular finite element grids and grid properties – User’s guide. Technical report, Waterloo Institute for Groundwater Research, University of Waterloo, Waterloo (1995). http://www.science.uwaterloo.ca/~mclaren/

  19. Merchant, M., Weatherill, N.: Adaptivity techniques for compressible inviscid flows. Comput. Methods Appl. Mech. Eng. 106(1–2), 83–106 (1993)

    Article  Google Scholar 

  20. 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)

    Google Scholar 

  21. Pinder, G.: Groundwater Modeling Using Geographical Information Systems. Wiley, New York (2002)

    Google Scholar 

  22. Pokrajac, D., Lazic, R.: An efficient algorithm for high accuracy particle tracking in finite elements. Adv. Water Resour. 25(4), 353–369 (2002)

    Article  Google Scholar 

  23. Pollock, D.: Semianalytical computation of path lines for finite-difference models. Groundwater 26(6), 743–750 (1988)

    Article  Google Scholar 

  24. Press, W., Flannery, B., Teukolsky, S., Vetterling, W.: Numerical Recipes in C. Cambridge University Press, Cambridge (1988)

    Google Scholar 

  25. Rannacher, R., Bangerth, W.: Adaptive Finite Element Methods for Differential Equations. Birkhäuser, Basel (2003)

    Google Scholar 

  26. Raper, J.: Three Dimensional Applications in Geographic Information Systems. Taylor and Francis, London (1989)

    Google Scholar 

  27. Sadek, E.: A scheme for the automatic generation of triangular finite elements. Int. J. Numer. Methods Eng. 15(12), 1813–1822 (1980)

    Article  Google Scholar 

  28. Shewchuk, J.: TRIANGLE: a two-dimensional quality mesh generator and Delaunay triangulator. Technical report, University of California, Computer Science Division, Berkeley (2005). https://www.cs.cmu.edu/~quake/triangle.html

  29. 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)

    Google Scholar 

  30. Thompson, J., Soni, B., Weatherill, N.: Handbook of Grid Generation. CRC, Boca Raton (1999)

    Google Scholar 

  31. Verfürth, R.: A Review of a Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques. Wiley/Teubner, New York/Stuttgart (1996)

    Google Scholar 

  32. Watson, D.: Computing n-dimensional Delaunay tesselation with application to Voronoi polytopes. Comput. J. 24(2), 167–172 (1981)

    Article  Google Scholar 

  33. Zheng, C., Bennett, G.: Applied Contaminant Transport Modeling: Theory and Practice. Van Nostrand Reinhold, New York (1995)

    Google Scholar 

  34. Zienkiewicz, O., Taylor, R.: The Finite Element Method. Volume 1: The Basis, 5th edn. Butterworth-Heinemann, Oxford (2000)

    Google Scholar 

  35. 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)

    Article  Google Scholar 

  36. 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)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Groundwater Modelling Centre DHI-WASY GmbH, Berlin, Germany

    Hans-Jörg G. Diersch

Authors
  1. Hans-Jörg G. Diersch
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

Copyright information

© 2014 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Diersch, HJ.G. (2014). Specific Topics. In: FEFLOW. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-38739-5_15

Download citation

Publish with us

Policies and ethics

Search

Navigation