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From the above observations, we can state that due to the nature of the shape functions of a FEM approximation, the exact solution of a simple non smooth problem cannot be obtained unless we have previously build a tailored conforming mesh. Moreover, it clearly appears that, to get rid of the requirement to place a node on the discontinuity, the polynomial approximation space has to be expanded such that it can reproduce exactly the discontinuity inside an element. In order to highlight the advantage of the PUM method over different enrichment methods, we first introduce the idea of a global enrichment using the simplest approach consisting in adding an enrichment function to the FEM approximation.

First, the expanded approximation space has to be constructed and an enrichment function has to be chosen. It is obvious that this enrichment function should own the same characteristics than the field in order to be able to render the exact solution. In the case of the bi-material rod, the response is piecewise linear, possesses a kink where the change of material occurs and has a discontinuous field derivative. As a first approach to reproduce the kink in the displacement response, one can simply add a piecewise linear function with a kink. In this case, one defines an enrichment function g(x) as:

g(x) =

 _x

l, 0 > x≤ l L−x

L−l, l > x≤ L

And a first enriched approximation can be given by: uh(x) =X

i∈I

Ni(x)ui+ ag(x)

The function g(x) spas over the domain [0,L]. It is therefore called a global enrichment function and the degree of freedom ’a’ has a global influence on the response.

We first model the bi-material with 3 elements of length L/3 while it will be shown later that one element is enough. The discretized field over the domain is given by:

0 0.2 0.4 0.6 0.8 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 X Displacement u(x)

Global Enrichment solution Analytical solution

Figure 3.4: Global-Local solution of the bi-material rod

We can notice in Figure 3.4 that this approximation can retrieve the exact solution of the bi- material rod. However, this approximation, which can be referred as the Global-Local Method of Mote [119], is in fact not very well suited in practice. Indeed the enrichment function is global, which means that while the center element is the sole element that needs a specific enriched approximation space, the whole domain is enriched. Hence, it is not very efficient as we have to integrate unnecessary terms on these elements. Moreover, it also means that this approach destroys the banded structure of the stiffness matrix. This problem can simply be solved by setting the function value of g(x) to be zero where the enrichment is not necessary. Hence, the enrichment becomes local and the stiffness matrix remains sparse. While it is not a problem in this 1D problem, a unique enrichment function and consequently a unique degree of freedom ’a’ generally introduces a lot of difficulties in higher dimensions because the construction of the enrichment function can become very arduous. Finally, this global enrichment does not possess the interpolant property of the FEM approximation as the displacement values at a node are not equal to u_iunless in the trivial case where ’a’ equal zero or if the g(x) function vanishes on nodes. To circumvent these problems Melenk and Babuska [110] proposed the PUM. In this approach a set of degrees of freedom a_i is introduced instead of a single parameter ’a’ and the enrichment function g(x) is multiplied by the term P

i∈I∗N_i∗(x)ai leading to the following approximation:

uh(x) =X i∈I uiNi(x) | {z } F EM +X i∈I∗ aiNi∗(x)g(x) | {z } Enrichment (3.1)

In [110] and [84] Melenk and Babuska developed the mathematical foundation of the PUM, demonstrating the efficiency of the method and including a proof of the convergence of PUM enriched approximation. A necessary condition to ensure the convergence requires that the functions Ni(x) and N_i∗(x) build a partition of unity over the analysis domain: P_i∈INi(x) = 1

and P

i∈INi∗(x) = 1. Thus, in the FEM, the functions Ni∗(x) and Ni(x) are chosen to be the

classical the shape functions, and generally N_i∗(x) and Ni(x) are the same functions yielding to:

uh(x) =X

i∈I

In this case, the sets of nodes I and I∗ are identical and the number of degrees of freedom can grow considerably with respect to the size of the standard FEM approximation as the problem size doubles.

In the seminal papers on PUM [84, 110], there is no assumption on the approximation method and thus the Ni(x) and Ni∗(x) can be any functions that satisfies the PU. The multiplication

by the N_i∗(x)g(x) plays a crucial role for the convergence property of the method, but it also creates a local enrichment that allows to keep a sparse system of equations since the N_i∗(x) shape functions are non-zero only on the compact support wi of node i. Notice that function g(x) is

multiplied by the standard shape functions Nj(x) for three reasons:

- we get the important property of partition of unity for the function g(x) becauseP

i∈I

Ni(x)=1

and the enrichment function can be represented exactly.

- if only one enrichment function g(x) is used for all the additional degrees of freedom (DOFs), the multiplication provides several N_j(x)g(x) which are independent for each DOF; - since the Ni(x) shape functions are non-zero only on the compact support wi of node i,

the system of equations remains sparse.

As one can notice in (3.1), in the most general case, the approximation does not have the Kronecker-δ property. Hence, uh(xi) 6= ui and the imposition of the essential boundary condi-

tions is difficult as we have to evaluate all terms of the approximation to correctly enforce the boundary condition. This happens when the enrichment function g(x) does not vanish on the element boundaries. Hence, by comparing the two different enrichments g₁(x) and g2(x) depicted

in Figure 3.5 (b), it is clear that it is preferable to use g1(x) as it vanishes on extremity nodes.

Using g₁(x) as an enrichment function, the imposition of essential boundary conditions is straight- forward and the solution obtained for the bi-material rod is depicted in Fig. 3.5 (a).

0 0.2 0.4 0.6 0.8 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 X Displacement u(x) GFEM solution Analytical solution

(a) GFEM solution of the bi-material rod

g (x)

g (x)

(b) GFEM enrichment for the bi-material rod

Figure 3.5: GFEM solution and enrichment function of the bi-material rod

the nodes where the boundary conditions are to be enforced. Thus, to recover the Kronecker- δ property on the whole domain, we have to construct carefully the enrichment function. A convenient procedure to obtain such enrichment is generally achieved by shifting the enrichment function by subtracting the nodal value g(xi) to the enrichment:

uh(x) =X i∈I uiNi(x) + X i∈I N_i∗(x)(g(x)_{− g(x}i))ai

This method was first proposed in [24], and it can be shown that this formulation can still reproduce the enrichment g(x). However, one has to note that the enrichment is now equal to zero on the nodes but it is different from zero along the element boundaries which can be a problem to enforce essential boundary conditions along the FE boundaries.