LBB condition ============= .. _OLE_LINK3: The approximations chosen on the one hand for the displacement and on the other hand for the contact pressures do not seem to satisfy the condition *inf-sup* in all cases. Failure to comply with condition LBB causes oscillations in contact pressures, a phenomenon comparable to that encountered in incompressibility [bib :ref:`13 `]. Physically, in the case of Lagrangian contact, this amounts to wanting to impose contact at too many points on the interface (over-stress), making the system hyperstatic. To release it, you have to restrict the space of the Lagrange multipliers, as is done in [bib :ref:`14 `] for Dirichlet conditions with X- FEM. The algorithm proposed by Moës [bib :ref:`14 `] to reduce oscillations is extended to the 3D case. Its aim is to impose equal relationships between Lagrange multipliers. This algorithm has been improved to make it more physical and more efficient. In the case of penalized contact, the same oscillations are found. For a triangle mesh, for example, we can show that there is no combination of Heaviside degrees of freedom that allows a rotation of the mean surface of the crack without generating oscillations of the displacement jump. For a high interface stiffness, as is the case in penalization, this generates pressure oscillations. To remedy this, it is necessary to recover the explicit pressure in the degree of freedom :math:`\lambda`, to apply the condition of LBB to it and to bring it back into balance, which explains the formulation given in § :ref:`3.3 `. The contact statuses are updated as in the Lagrangian case, where :math:`\lambda` and not :math:`{d}_{n}` are tested to go from a contacting state to a non-contacting state, in order to avoid oscillations in the contact statuses. Description of the Moës' algorithm for linear and quadratic elements (algo1) ------------------------------------------------------------------------ The algorithm introduced by Moës [bib :ref:`14 `] is presented there in order to impose Dirichlet conditions on an interface as part of X- FEM. It shows that the Lagrange multiplier technique to impose Dirichlet conditions must be used carefully, because the*inf-sup* condition is not always respected. The paper is restricted to the 2D case, but the algorithm presented is easily generalizable to the 3D case. The first phase is a node selection phase, in which the nodes selected are the ones that are "important" for approximating the Lagrange multipliers. The other nodes are overabundant and lead to oscillations in the Lagrange multipliers. Once the "important" nodes have been selected, equality relationships are imposed between the Lagrange multipliers, in order to restrict the space of the multipliers. Thus, the Lagrange multipliers of the edges emanating from the same selected node are equal. More formally, :math:`E` and :math:`N` are the sets containing all the edges and nodes of the mesh. The two ends of an edge :math:`e\in E` are marked :math:`({v}_{1}(e),{v}_{2}(e))\in {N}^{2}`. First, we start with an initialization phase (iteration :math:`k=0` of the algorithm). First of all, we determine :math:`{S}_{e}^{0}`, the set of edges that are cut by the interface. Since the interface is represented by the normal level set :math:`\text{lsn}`, an edge :math:`e\in E` is cut by the interface if and only if :math:`\text{lsn}({v}_{1}(e))\cdot \text{lsn}({v}_{2}(e))\le 0`. Note that if the interface coincides with node :math:`{v}_{1}(e)` or node :math:`{v}_{2}(e)`, edge :math:`e` is owned by :math:`{S}_{e}^{0}`: :math:`{S}_{e}^{0}=\left\{e\in E,\text{lsn}({v}_{1}(e))\cdot \text{lsn}({v}_{2}(e))\le 0\right\}` Let :math:`{N}_{e}` be the set of nodes connected by the elements of :math:`{S}_{e}^{0}`: :math:`{N}_{e}=\left\{n\in N,\exists e\in {S}_{e}^{0}n={v}_{1}(e)\text{ou}n={v}_{2}(e)\right\}` Let :math:`{S}_{n}^{0}` be the set of nodes selected in iteration :math:`k=0` (initialization). These nodes are the ones that coincide with the interface (this set may be empty): :math:`{S}_{n}^{0}=\left\{n\in {N}_{e},\text{lsn}(n)=0\right\}` After this initialization phase, the algorithm iterates from :math:`k=\mathrm{1,}\text{nmax\_iter}`. At each iteration, the following steps are carried out: * Update all of the edges: we delete those that are connected to a node selected in the previous iteration .. _OLE_LINK4: :math:`{S}_{e}^{k}={S}_{e}^{k-1}\setminus \left\{e\in {S}_{e}^{k-1},{v}_{1}(e)\in {S}_{n}^{k-1}\text{ou}{v}_{2}(e)\in {S}_{n}^{k-1}\right\}` * Calculating the node score: for each node in :math:`{N}_{e}`, we calculate a score composed of 2 digits: the first number corresponds to the number of connected edges in :math:`{S}_{e}^{k}`, and the second corresponds to the absolute value of the normal level set for this node. This score :math:`\text{sc\_no}` is therefore a two-column matrix whose rows represent the node. :math:`\forall n\in {N}_{e}\{\begin{array}{}{\text{sc\_no}}^{k}(n\mathrm{,1})=\text{nombre}\text{d'arêtes}\text{connectées}\text{au}\text{noeud}n\\ {\text{sc\_no}}^{k}(n\mathrm{,2})=\mid \text{lsn}(n)\mid \end{array}` * Calculation of the edge score: for each edge in :math:`{S}_{e}^{k}`, we calculate a score composed of 2 digits: the first number corresponds to the absolute value of the difference of the 1st digit of the score of the 2 end nodes, and the second corresponds to a ratio between the values of the 2nd number of the 2 end nodes (i.e. a value ratio of :math:`\text{lsn}`. This score :math:`\text{sc\_ar}` is therefore a matrix with two columns whose rows represent the edge. :math:`\begin{array}{}\forall e\in {S}_{e}^{k},\forall j\in \left\{\mathrm{1,2}\right\},{s}_{j}={\text{sc\_no}}^{k}({v}_{j}(e)\mathrm{,1}),{l}_{j}={\text{sc\_no}}^{k}({v}_{j}(e)\mathrm{,2})\\ \{\begin{array}{}{\text{sc\_ar}}^{k}(e\mathrm{,1})=\mid {s}_{1}-{s}_{2}\mid \\ {\text{sc\_ar}}^{k}(e\mathrm{,2})=\{\begin{array}{cc}\frac{{l}_{1}}{{l}_{1}+{l}_{2}}& \text{si}{s}_{1}<{s}_{2}\\ \frac{{l}_{1}}{{l}_{1}+{l}_{2}}& \text{si}{s}_{1}>{s}_{2}\\ \frac{\text{min}({l}_{\mathrm{1,}}{l}_{2})}{{l}_{1}+{l}_{2}}& \text{si}{s}_{1}={s}_{2}\end{array}\end{array}\end{array}` * Search for the "best edge" :math:`{b}_{e}`: it is the edge whose 1st number of its score is the largest. In case of a tie between 2 edges, it is the one whose 2nd number of its score is the largest. * Search for the "best node" :math:`{b}_{n}`: it is the end node of :math:`{b}_{e}` whose 1st number of its score is the largest. In case of a tie, it is the node whose 2nd number of the score is the lowest (the node closest to the interface). Node :math:`{b}_{n}` is the only node selected at this iteration: :math:`{S}_{n}^{k}=\left\{{b}_{n}\right\}` The algorithm stops if during an iteration, set :math:`{S}_{e}^{k}` becomes the empty set. The final set of selected nodes will then be: :math:`W=\underset{k}{\cup }{S}_{n}^{k}` After this node selection phase, the algorithm builds the Lagrange multiplier space, whose size is equal to that of :math:`W`. So the multiplier space is: :math:`{S}_{\lambda }=\left\{{\lambda }^{i},i\in \left\{\mathrm{1,}\text{card}(W)\right\}\right\}` With this algorithm any node in :math:`{N}_{e}\text{/}W` is connected by an edge of :math:`{S}_{e}^{0}` to a node in :math:`W`. An equal relationship is then imposed between these two nodes. In case of conflict (connections to several nodes in :math:`W` by as many edges), the nodes in :math:`W` are discriminated against by criterion 2 of the nodes (lowest normal level-set). Note that this algorithm can also be used in large swings. Description of the modified algorithm for linear and quadratic elements (algo2) ------------------------------------------------------------------------ Based on similar ideas, a new algorithm was proposed. Thus, in the new version, we start from all the edges on which the normal level set is cancelled at least at one point. These edges connect points on both sides of the interface (or possibly points on the interface). The algorithm looks for the minimum subset of edges that can connect all the end points of the edges. Then, groups of connected edges are extracted from it. The relationships imposed are then as follows: * the multipliers on the top nodes of each group are imposed equal, More formally, :math:`E` and :math:`N` are the sets containing all the edges and nodes of the mesh. The two ends of an edge :math:`e\in E` are marked :math:`({v}_{1}(e),{v}_{2}(e))\in {N}^{2}`. First of all, we determine :math:`{S}_{e}`, the set of edges that are strictly cut by the interface. Since the interface is represented by the normal level set :math:`\text{lsn}`, an edge :math:`e\in E` is strictly cut by the interface if and only if :math:`\text{lsn}({v}_{1}(e))\cdot \text{lsn}({v}_{2}(e))<0`. Note that if the interface coincides with node :math:`{v}_{1}(e)` or node :math:`{v}_{2}(e)`, edge :math:`e` is not in :math:`{S}_{e}`: :math:`{S}_{e}=\left\{e\in E,\text{lsn}({v}_{1}(e))\cdot \text{lsn}({v}_{2}(e))<0\right\}` Let :math:`{N}_{e}` be the set of nodes connected by the elements of :math:`{S}_{e}`. We separate :math:`{N}_{e}` into two parts: the nodes "under" and "above" the crack, according to the sign of :math:`\text{lsn}`: :math:`\begin{array}{}{N}_{e}=\left\{n\in N,\exists e\in {S}_{e}n={v}_{1}(e)\text{ou}n={v}_{2}(e)\right\}\\ {N}_{e}^{+}=\left\{n\in {N}_{e},\text{lsn}(n)>0\right\}\text{et}{N}_{e}^{-}=\left\{n\in {N}_{e},\text{lsn}(n)<0\right\}\end{array}` The algorithm is looking for :math:`{S}_{\text{ve}}`, the minimal subset of :math:`{S}_{e}` that connects nodes in :math:`{N}_{e}^{\text{+}}` to nodes in :math:`{N}_{e}^{-}`. Each node in :math:`{N}_{e}^{\text{+}}` must be connected to at least one node in :math:`{N}_{e}^{-}` s, and each node in :math:`{N}_{e}^{-}` must be connected to at least one node in :math:`{N}_{e}^{\text{+}}`. The edges in :math:`{S}_{\text{ve}}` are called "vital edges" because if one of these edges disappears, at least one node in :math:`{N}_{e}` will be orphaned. This set of vital edges is not necessarily unique. In the presence of choices, the shortest vital edge is preferred. As will be seen later, this amounts to minimizing the approximation region P 0. For the search for set :math:`{S}_{\text{ve}}`, we chose an algorithm based on the notion of node and edge scores, a concept found in algo1. The algorithm will remove all non-vital edges one by one, until only vital edges are left. More precisely, a score is associated with each node, which corresponds to the number of edges connected to this node. Each edge is associated with a score, which corresponds to the minimum of the scores of the two end nodes. Let :math:`e` be the edge with the highest score (with the same score, the longest edge is preferred). If the score of :math:`e` is 1, then any edges that are left are vital edges. :math:`{S}_{\text{ve}}` is determined and the algorithm stops. If the score of :math:`e` is strictly greater than 1, edge :math:`e` is a non-vital edge, and it is symbolically removed from the list of edges :math:`{S}_{e}`. The algorithm starts again, recalculating the node score, and so on until only vital edges are left. It's important to note that :math:`{S}_{\text{ve}}` is made up of some disconnected edges, and some edges that are connected to each other. These groups of connected vital edges are taken from :math:`{S}_{\text{ve}}`. Note that in such a group, all the edges are connected by a single node (see :ref:`Figure 6.2-1 `). Let :math:`{G}_{\text{cve}}^{i}` be the group of vital edges connected by node :math:`i`. So :math:`{G}_{\text{cve}}^{i}` is defined by: :math:`{G}_{\text{cve}}^{i}=\left\{e\in {S}_{\text{ve}},i={v}_{1}(e)\text{ou}i={v}_{2}(e)\right\}` Now, relationships are imposed between Lagrange multipliers. All multipliers carried by the top nodes in the same group are taxed equal. We illustrate these algorithms on the 2D case of :ref:`Figure 6.2-1 `. Node groups linked by relationships equal to version 1 are marked by blue circles. Node groups linked by relationships equal to version 2 are marked by solid edges connected to each other. Note that this algorithm can also be used in large swings. .. image:: images/10000000000002A50000016A5B8F064A5C0D0145.png :width: 4.8083in :height: 2.3575in .. _RefImage_10000000000002A50000016A5B8F064A5C0D0145.png: .. _Ref130982253: **Figure** 6.2-1 **: Example of edges cut by an interface and the resulting approximation** Description of the algorithm for quadratic elements (algo 3): --------------------------------------------------------- This algorithm makes it possible to impose the minimum number of relationships of equality in order to reach the optimal convergence rate for quadratic elements. The determination of the edges carrying the relationships of equality between the "Lagranges" takes place in two stages: * Study of related components: A connected component is a set of cut edges linked together continuously (by one of the two vertices of each edge). The algorithm analyzes all the cut edges in order to determine the related components, then it puts an equality relationship on the first edge of each component. .. image:: images/100000000000032A00000111A8400A17CC1A6204.jpg :width: 3.7134in :height: 1.039in .. _RefImage_100000000000032A00000111A8400A17CC1A6204.jpg: Figure 6.3-1: Equality relationships on related components .. image:: images/100000000000017B000000A636FA291E6E9A2A2A.jpg :width: 1.9835in :height: 0.8547in .. _RefImage_100000000000017B000000A636FA291E6E9A2A2A.jpg: * Study of the proximity criterion: The aim is to analyze, for each cut edge, the proximity of the nodes to the interface. The idea is to link the "Lagrange" nodes with "weak" influence. Concretely, we start by eliminating the non-vital edges in order to end up with a subset of cut edges connecting all the nodes on both sides of the interface. Then the equality relationships are put on each edge where the proximity criterion is verified. Finally, we re-study the non-vital edges: if one of them connects two edges, only one of which verifies the proximity criterion, the non-vital edge in question bears a tie; in the other cases nothing is done. .. image:: images/10000000000003680000012626F9D49C6309B81B.jpg :width: 4.1618in :height: 1.1917in .. _RefImage_10000000000003680000012626F9D49C6309B81B.jpg: Figure 6.3-2: Selecting the group of vital edges .. image:: images/100000000000019C000000A02E58F7C2F757C8B4.jpg :width: 2.2035in :height: 0.9646in .. _RefImage_100000000000019C000000A02E58F7C2F757C8B4.jpg: .. image:: images/100000000000034F00000125431D7D93BA7E44AF.jpg :width: 3.8091in :height: 1.1264in .. _RefImage_100000000000034F00000125431D7D93BA7E44AF.jpg: Figure 6.3-3: Equality relationships on related components and proximity criteria .. image:: images/100000000000024500000072F539EE437F8C635F.jpg :width: 2.448in :height: 0.5165in .. _RefImage_100000000000024500000072F539EE437F8C635F.jpg: More formally, and using the notations used in §6.2, we consider :math:`{S}_{\text{e}}` to be the set of edges strictly cut by the interface. The first step is to partition Se into a union of disjoint related components: :math:`{S}_{\text{e}}={U}_{\text{i}}{S}_{\text{ei}}` An equality relationship is imposed on the first edge of each subset :math:`{S}_{\text{ei}}`. Next, we build :math:`{S}_{\text{ve}}`, the group of "vital" edges (in the sense of algo 2), to study the proximity criterion. :math:`{S}_{\text{ve}}` is composed of two types of edges: independent edges, and edges connected to each other via a common vertex. We denote :math:`{S}_{\text{ind}}` and :math:`{S}_{\text{cte}}` the subsets of :math:`{S}_{\text{ve}}` corresponding to these two types respectively, such as: :math:`{S}_{\text{e}}={S}_{\text{ind}}U{S}_{\text{cte}}` The proximity criterion is studied on the following nodes: * for the edges of :math:`{S}_{\text{ind}}` we look for the node closest to the interface and if the distance (relative) is less than the threshold distance set by the criterion, the edge has an equality relationship; * for the edges of :math:`{S}_{\text{cte}}` we calculate the minimum distance from the common vertex, and if this minimum distance is less than the threshold we impose equal relationships on all the edges of the packet. In this way we manage to link the "Lagranges" with "weak" influence. Note that this algorithm cannot be used in the context of large slides, the various possible pairings changing as the sliding occurs. .. _Ref143502469: Imposed relationships between friction semi-multipliers ---------------------------------------------------------------- When friction is taken into account, the same phenomenon of oscillations is observed on friction semi-multipliers as in the previous case on contact multipliers. In order for the contact reaction to no longer oscillate, it is necessary to eliminate the oscillations of the normal reaction (contact pressure) and of the tangential reaction (therefore friction semi-multipliers). To do this, it is therefore also necessary to activate the algorithm for restricting multiplier spaces for friction semi-multipliers. The relationships are to be imposed on tangential reactions, and involve the friction unknowns :math:`{\Lambda }_{1}` and :math:`{\Lambda }_{2}` as well as the covariant base vectors :math:`{\tau }_{1}` and :math:`{\tau }_{2}`. For the case of imposing an equal relationship between nodes :math:`A` and :math:`B`, the relationship is written as: :math:`{\Lambda }_{1}^{A}{\tau }_{1}^{A}+{\Lambda }_{2}^{A}{\tau }_{2}^{A}={\Lambda }_{1}^{B}{\tau }_{1}^{B}+{\Lambda }_{2}^{B}{\tau }_{2}^{B}` Since the two unknowns to be determined (:math:`{\Lambda }_{1}^{A}` and :math:`{\Lambda }_{2}^{A}` for example) are scalar, it is necessary to transform the previous vector relationship into two scalar relationships. This is done by projecting onto the :math:`({\tau }_{1}^{A},{\tau }_{2}^{A})` base. The two relationships to be imposed are therefore finally: .. _RefEquation 6.4-1: :math:`\{\begin{array}{}{\Lambda }_{1}^{A}({\tau }_{1}^{A}\cdot {\tau }_{1}^{A})+{\Lambda }_{2}^{A}({\tau }_{2}^{A}\cdot {\tau }_{1}^{A})={\Lambda }_{1}^{B}({\tau }_{1}^{B}\cdot {\tau }_{1}^{A})+{\Lambda }_{2}^{B}({\tau }_{2}^{B}\cdot {\tau }_{1}^{A})\\ {\Lambda }_{1}^{A}({\tau }_{1}^{A}\cdot {\tau }_{2}^{A})+{\Lambda }_{2}^{A}({\tau }_{2}^{A}\cdot {\tau }_{2}^{A})={\Lambda }_{1}^{B}({\tau }_{1}^{B}\cdot {\tau }_{2}^{A})+{\Lambda }_{2}^{B}({\tau }_{2}^{B}\cdot {\tau }_{2}^{A})\end{array}` eq 6.4-1 This choice is called into question by the introduction of major slippage [:external:ref:`R5.03.53 `]: in fact the contact base changes with each geometric iteration. The previous relationships introduced durably using the initial base therefore no longer make sense. To resolve the conflict, we instead introduce the two relationships of equality on the components: .. _RefEquation 6.4-2: :math:`\{\begin{array}{}{\Lambda }_{1}^{A}={\Lambda }_{1}^{B}\\ {\Lambda }_{2}^{A}={\Lambda }_{2}^{B}\end{array}` eq 6.4-2 Considering that the bases :math:`({\tau }_{1}^{A},{\tau }_{2}^{A})` and :math:`({\tau }_{1}^{B},{\tau }_{2}^{B})` are almost identical due to the proximity of the points :math:`A` and :math:`B`, the equations and are almost equivalent. Notes on the relationships imposed by algorithm 1 or 2 ----------------------------- On contact multipliers ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Consider the linear relationship between contact Lagrange multipliers :math:`{\lambda }_{1}`, :math:`{\lambda }_{2}`, and :math:`{\lambda }_{3}`: :math:`{\lambda }_{2}={\text{αλ}}_{1}+(1-\alpha ){\lambda }_{3}` The relationship relates to the pressure value and not to the contact pressure vector. In the case of a curved structure, a relationship on pressure vectors of the type: .. code-block:: text :math:`{\lambda }_{2}{n}_{2}={\text{αλ}}_{1}{n}_{1}+(1-\alpha ){\lambda }_{3}{n}_{3}` is not necessarily possible because vector :math:`{n}_{2}` is not an unknown. On friction semi-multipliers ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ The unknown for friction is vectorial. One could imagine linking the tangent reaction vectors together: :math:`{r}_{{\tau }_{2}}={\mathrm{\alpha r}}_{\text{τ1}}+(1-\alpha ){r}_{\text{τ3}}` .. code-block:: text gold :math:`{r}_{{\tau }_{i}}={\text{μλ}}_{i}{\Lambda }_{i}` What gives :math:`{\Lambda }_{2}=\frac{\alpha {\lambda }_{1}{\Lambda }_{1}+(1-\alpha ){\lambda }_{3}{\Lambda }_{3}}{\alpha {\lambda }_{1}+(1-\alpha ){\lambda }_{3}}=\beta {\Lambda }_{1}+(1-\beta ){\Lambda }_{3}` with :math:`\beta =\frac{\alpha {\lambda }_{1}}{\alpha {\lambda }_{1}+(1-\alpha ){\lambda }_{3}}\approx \alpha` if the mesh is fine enough :ref:`Figure 6.5.2-1 ` This choice of relationship is impossible because in 3D, if points 1 and 3 are in sliding contact, then point 2 will not be! Indeed, the norm of :math:`{\Lambda }_{2}` will be strictly less than 1 if the sliding directions are not collinear (see). .. image:: images/10000000000001e2000000d79d83 EAA11F92E89A .png :width: 2.2591 in :height: 1.0299 in .. _refImage_10000000000001e2000000d79d83 EAA11F92E89A .png: .. _Ref148872427: **Figure** 6.5.2-1 **: Case of an adhering point between two sliding points** .. code-block:: text The proposed solution is to impose a linear relationship between the norms of the tangent reaction vectors, which is equivalent to imposing a relationship between the norms of the friction semi-multipliers: :math:`\parallel {r}_{\text{τ2}}\parallel =\alpha \parallel {r}_{\mathrm{\tau 1}}\parallel +(1-\alpha )\parallel {r}_{\mathrm{\tau 3}}\parallel \iff \parallel {\Lambda }_{2}\parallel =\beta \parallel {\Lambda }_{1}\parallel +(1-\beta )\parallel {\Lambda }_{3}\parallel` eq. 6.5.2-1 .. code-block:: text This relationship is non-linear due to the norm. Newton's method allows us to reduce ourselves to the successive imposition of linear relationships. At Newton's iteration I, the relationship is: :math:`{\mathrm{\delta \Lambda }}_{2}^{i}\text{.}\frac{{\Lambda }_{2}^{i-1}}{\parallel {\Lambda }_{2}^{i-1}\parallel }-\text{βδ}{\Lambda }_{1}^{i}\text{.}\frac{{\Lambda }_{1}^{i-1}}{\parallel {\Lambda }_{1}^{i-1}\parallel }-(1-\beta ){\mathrm{\delta \Lambda }}_{3}^{i}\text{.}\frac{{\Lambda }_{3}^{i-1}}{\parallel {\Lambda }_{3}^{i-1}\parallel }=\beta \parallel {\Lambda }_{1}^{i-1}\parallel +(1-\beta )\parallel {\Lambda }_{3}^{i-1}\parallel -\parallel {\Lambda }_{2}^{i-1}\parallel` This type of linear relationship is currently not available in Code_Aster, where only linear relationships whose coefficients are constant for the duration of the calculation are allowed. Currently, the relationship between implanted friction semi-multipliers is as follows: :math:`{\Lambda }_{2}={\mathrm{\alpha \Lambda }}_{1}+(1-\alpha ){\Lambda }_{3}` When the sliding or the grip are unidirectional, we find the equation [:external:ref:`éq. 4.6-1 <éq. 4.6-1>`] by substituting :math:`\alpha` for :math:`\beta`. .. _RefNumPara__33433431: Non-simplicial elements (quadrangles, hexahedra...) ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ .. code-block:: text In the case of 2D quaden, penta or hexagonal 3D meshes, certain configurations are noted where nodes of the mesh do not belong to a cut edge. In the two examples in the figure, the red knots are not connected to any other knot. In order to satisfy LBB, the idea is to connect these nodes to the related nodes but try not to introduce linear relationships in this case. .. image:: images/Object_1518.png :width: 5.9984in :height: 2.3339in .. _RefSchema_Object_1518.png: .. _Ref1488724271: Figure 6.5.3-1: Red knots do not belong to a cut edge. .. code-block:: text To avoid adding linear relationships, the idea is to eliminate nodes from the interpolation that don't belong to a cut edge. To satisfy the partition of the unit of contact contributions, the shape functions of the eliminated nodes are distributed over the other nodes using the following uniform distribution: :math:`\tilde{\phi }{\text{}}_{i\in {N}_{\mathrm{actif}}}={\phi }_{i}+{\sum }_{j\in {N}_{\mathrm{elim}}}{\phi }_{j}/{N}_{\mathrm{actif}}` eq. 6.5.3-1 Where :math:`{N}_{\mathrm{actif}}` is the set of nodes in the element that are directly cut or belong to a cut edge. :math:`{N}_{\mathrm{elim}}` is the set of knots to be eliminated. For the quadrangle in the figure, the modified shape functions are then written :math:`\tilde{\phi }{\text{}}_{i=\mathrm{1,2}\mathrm{,4}}={\phi }_{i}+{\phi }_{3}/3`. For the hexahedron they are written :math:`\tilde{\phi }{\text{}}_{i=\mathrm{1,2}\mathrm{,4}\mathrm{,5}\mathrm{,6}\mathrm{,7}\mathrm{,8}}={\phi }_{i}+{\phi }_{3}/7`. We note that other distribution choices are possible: for example, we could have chosen to distribute an eliminated node over its associated active nodes. For the quadrangle in the figure, the modified shape functions would then be written :math:`\tilde{{\phi }_{1}}={\phi }_{1}` and :math:`\tilde{\phi }{\text{}}_{i=\mathrm{2,4}}={\phi }_{i}+{\phi }_{3}/2`. :ref:`4.4 ` The elimination of excess contact degrees of freedom is discussed [§]), we choose substitution (by putting 1 on the diagonal and 0 in the second member). :ref:`4.3.2 ` This approach can be generalized for crack bottoms. Indeed, in the two examples in the figure, the edge {1-4} is cut while {2-3} is not cut: the nodes 2 and 3 are therefore to be eliminated. Using the approach described earlier, modified form functions are written :math:`\tilde{\phi }{\text{}}_{i=\mathrm{1,4}}={\phi }_{i}+({\phi }_{2}+{\phi }_{3})/2`. This is equivalent to doing the P0 integration described in paragraph [§] on the cut elements containing the tip. .. image:: images/Object_1526.png :width: 4.5437 in :height: 1.6453 in .. _RefSchema_Object_1526.png: .. _Ref14887242711: **Figure** 6.5.3-2 **: Red knots do not belong to a cut edge.** .. code-block:: text