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Mathematical background for X-FEM

tuhinsinha.25's picture


I am a graduate student at Purdue University working in the area of powder mechanics. I have recenlty come accross the field of extended FEM and have begun to underastand its potential advantage over commercial FEM in solving problems involving singularities, arbitrary discontinuties and problems inovlving lots of remeshing or mesh distrotions (finite plasticity problems). I am typically interested in the last application of X-FEM and would like to study it in more detail. I started off by reading some prelimnary papers by Babuska et al., Belytschko et al. etc. but I lack the preliminary mathematical background to understand the mathematical analysis involved in such problems (like partition of unity, basic definitions and operations in functional spaces for e.g. Lipschitz Space, Sobolev space etc.)

Can the experts in this field please point me out to good references, courses etc. that I should take in order to understand the underlying principles more comprehensively.





Take a course in real analysis followed by functional analysis in your math department.

At the same time, you can read any book related to analysis to develop the background.


N. Sukumar's picture


The very basics of functional analysis (vector spaces, norms, etc.) would suffice to develop an understanding of the math required in basis-set approaches such as FEM/X-FEM. The best book for FEM and also for X-FEM/PUFEM in this regard is: Strang and Fix ("An Analysis of the Finite Element Method", Prentice-Hall, 1973). It provides the essentials on the math for one (without a math degree) to readily understand and appreciate FEM, and still remains a classic in the field.  Most FEM textbooks quickly get into the implementation of the method (shape functions, B matrix); S&F provide the mathematical basis of the FEM, as the title indicates, which would aid in understanding partition-of-unity FE methods. Hope this helps?

tuhinsinha.25's picture

Thanks a lot Dr.Sukumar..I'll try that book and see how it goes..

I know the question was what material to read in order to understand the X-FEM method; however, it may also help to know what aspects of powder mechanics you are researching to determine whether X-FEM is the most appropriate method to use. Powder mechanics can be especially sensitive to the appropriate RVE for continuum methods, and many dense, dynamic powder systems are difficult to characterize using such approaches. 

Depending on the application, a host of other numerical methods may help to study your problem of interest. Some popular approaches include:


  1. multi-phase Eulerian hydrocodes (e.g., the open source MFIX at NETL)
  2. meshless Lagrangian continuum methods (e.g., Element-Free Galerkin (EFG) or Smoothed Particle Hydrodynamics (SPH))
  3. discontinuum methods (e.g., hard sphere particle dynamics or Discrete Element Methods (DEM))



The following book is great as an introduction to functional analysis and its application to variational methods.

D. Reddy, Introductory Functional Analysis: With Applications to Boundary Value Problems and Finite Elements



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