p.

p1 {margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Helvetica}p.

p2 {margin: 0.0px 0.0px 0.0px 0.0px; font: 9.

0px Helvetica}p.p3 {margin: 0.0px 0.0px 0.0px 0.0px; font: 17.0px Helvetica}span.s1 {font: 9.

0px Helvetica}span.s2 {font: 12.0px Helvetica}span.

s3 {font: 17.0px Helvetica}span.s4 {font: 7.

0px Helvetica}Equation (1.4) generally defines the XFEM. For a particular realization of theXFEM, the choice of the nodal subset I?, global enrichment function r(x), and thepartition of unity functions N?i (x) has to be defined.The displacement approximation given by Equation (1.4) is called ‘extrinsic’ global enrichment(i.e., FE approximation basis is augmented with additional functions and allthe nodes in the FE mesh are enriched with r(x)).

This does not satisfy the Kronecker-d property (i.e., Ni(xj) =di j) which renders the imposition of essential boundary conditionsand the interpretation of results difficult, expect for the phantom node method60, 54. In most cases, the region of interest is localized, for example, cracks ormaterialinterfaces and hence the enrichment could be restricted closer to the regionof interest. This type of enrichment is called ‘local enrichment’. Moreover, a globalenrichment is computationally demanding because the number of degrees of freedomis proportional to the number of nodes and the number of enrichment functions andthe resulting system matrix is not banded. A ‘shifted enrichment’ is used to retain theKronecker-d property, given by:uh(x) = uhf em(x)+uh enr(x)=åi ?INi(x)ui+ åj?I?Nj(x)r(x)?r(xj)aj.

(1.5)where r(xj) is the value of the enrichment function evaluated at node j. An examplewill be considered of this shifting for 1D when r(x)= |f (x)|=|x?xb|, where xb is thelocation of the interface from the left end. Different enrichment functions are proposedin the literature to capture strong and weak discontinuities arising in different problems.For weak and strong discontinuities, the nodal subset I? is built from all nodes ofelements that are cut by the discontinuity.

Whether or not an element is cut by thediscontinuity can conveniently be determined on element-level by help of the level-setfunction f (x)???cut elements mini?Ielfi.maxi?Ielfi < 0uncut elements mini?Ielfi.maxi?Ielfi < 0.For weak discontinuities, where a solution shows a kink, or in other words, a jumpin the gradient, the global enrichment function is typically chosen as the abs-functionof the level-set function,r(x) = |f (x)|Along strong discontinuities, a jump is present in the solution. A typical choice for theglobal enrichment function is the sign-function (or Heaviside-function) of the level-setfunction,r(x) = signf (x) =????1 f (x) < 0,0 f (x) = 0,1 f (x) > 0