Journal of Shanghai Jiaotong University (Science)

, Volume 23, Issue 1, pp 158–165

# Mapped Displacement Discontinuity Method: Numerical Implementation and Analysis for Crack Problems

Article

## Abstract

The displacement discontinuity method (DDM) is a kind of boundary element method aiming at modeling two-dimensional linear elastic crack problems. The singularity around the crack tip prevents the DDM from optimally converging when the basis functions are polynomials of first order or higher. To overcome this issue, enlightened by the mapped finite element method (FEM) proposed in Ref. [13], we present an optimally convergent mapped DDM in this work, called the mapped DDM (MDDM). It is essentially based on approximating a much smoother function obtained by reformulating the problem with an appropriate auxiliary map. Two numerical examples of crack problems are presented in comparison with the conventional DDM. The results show that the proposed method improves the accuracy of the DDM; moreover, it yields an optimal convergence rate for quadratic interpolating polynomials.

### Key words

displacement discontinuity method (DDM) singularity auxiliary map convergence rate Hadamard finite part

O 241.82

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### References

1. [1]
VANDAMME L, DETOURNAY E, CHENG A H D. A twodimensional poroelastic displacement discontinuity method for hydraulic fracture simulation [J]. International Journal for Numerical and Analytical Methods in Geomechanics, 1989, 13(2): 215–224.
2. [2]
VERDE A, GHASSEMI A. A fast multipole displacement discontinuity method (FM-DDM) for geomechanics reservoir simulations [J]. International Journal for Numerical and Analytical Methods in Geomechanics, 2015, 39(18): 1953–1974.
3. [3]
TAO Q, EHLIG-ECONOMIDES C A, GHASSEMI A, et al. Investigation of stress-dependent fracture permeability in naturally fractured reservoirs using a fully coupled poroelastic displacement discontinuity model [C]//SPE Annual Technical Conference and Exhibition. New Orleans, Louisiana: Society of Petroleum Engineers, 2009: 1–4.Google Scholar
4. [4]
HENSHELL R D, SHAW K G. Crack tip finite elements are unnecessary [J]. International Journal for Numerical Methods in Engineering, 1975, 9(3): 495–507.
5. [5]
BARSOUM R S. On the use of isoparametric finite elements in linear fracture mechanics [J]. International Journal for Numerical Methods in Engineering, 1976, 10(1): 25–37.
6. [6]
CARSTENSEN C, STEPHAN E P. A posteriori error estimates for boundary element methods [J]. Mathematics of Computation, 1995, 64(210): 215–224.
7. [7]
GUO B Q, HEUER N. The optimal rate of convergence of the p-version of the boundary element method in two dimensions [J]. Numerische Mathematik, 2004, 98(3): 499–538.
8. [8]
GUO B Q, HEUER N. The optimal convergence of the h-p version of the boundary element method with quasiuniform meshes for elliptic problems on polygonal domains [J]. Advances in Computational Mathematics, 2006, 24(1): 353–374.
9. [9]
HOLM H, MAISCHAK M, STEPHAN E P. Exponential convergence of the h-p version BEM for mixed boundary value problems on polyhedrons [J]. Mathematical Methods in the Applied Sciences, 2008, 31(17): 2069–2093.
10. [10]
OH H S, BABUŠKA I. The method of auxiliary mapping for the finite element solutions of elasticity problems containing singularities [J]. Journal of Computational Physics, 1995, 121(2): 193–212.
11. [11]
OH H S, DAVIS C, KIM J G, et al. Reproducing polynomial particle methods for boundary integral equations [J]. Computational Mechanics, 2011, 48(1): 27–45.
12. [12]
OH H S, JEONG J W, KIM J G. The reproducing singularity particle shape functions for problems containing singularities [J]. Computational Mechanics, 2007, 41(1): 135–157.
13. [13]
CHIARAMONTE M M, SHEN Y, LEW A J. Mapped finite element methods: High-order approximations of problems on domains with cracks and corners [J]. International Journal for Numerical Methods in Engineering, 2017, 111(9): 864–900.
14. [14]
CROUCH S L. Solution of plane elasticity problems by the displacement discontinuity method. I. Infinite body solution [J]. International Journal for Numerical Methods in Engineering, 1976, 10(2): 301–343.
15. [15]
CRUSE T A. Boundary element analysis in computational fracture mechanics [M]. Dordrecht: Kluwer Academic Publishers, 1988.
16. [16]
HADAMARD J. Lectures on Cauchy’s problem in linear partial differential equations [M]. New Haven: Yale University Press, 1925.Google Scholar
17. [17]
GONNET P. Adaptive quadrature re-revisited [D]. Zurich: Department of Computer Science, Swiss Federal Institute of Technology, 2009.Google Scholar
18. [18]
ATTEWELL P B, FARMER I W. Principles of engineering geology [M]. New York: John Wiley & Sons Inc., 1976.
19. [19]
SNEDDON I N, Lowengrub M. Crack problems in the classical theory of elasticity [M]. New York: Wiley, 1969.
20. [20]
WEERTMAN J. Dislocation based fracture mechanics [M]. Singapore: World Scientific Publishing Co. Inc., 1996.

© Shanghai Jiaotong University and Springer-Verlag GmbH Germany, part of Springer Nature 2018

## Authors and Affiliations

• Feng Jiang (姜 锋)
• 1
• 2
• Yongxing Shen (沈泳星)
• 1
• 2
1. 1.State Key Laboratory of Metal Matrix CompositesShanghai Jiao Tong UniversityShanghaiChina
2. 2.University of Michigan - Shanghai Jiao Tong University Joint InstituteShanghai Jiao Tong UniversityShanghaiChina