김경종
477 lines
30 KiB
Markdown
477 lines
30 KiB
Markdown
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Before developing the inf-sup condition, let us state the ellipticity condition for the problem of total incompressibility: there is a constant $\alpha$ greater than zero and independent of h such that
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$$
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a (\mathbf {v} _ {h}, \mathbf {v} _ {h}) \geq \alpha \| \mathbf {v} _ {h} \| ^ {2} \quad \forall \mathbf {v} _ {h} \in K _ {h} (0) \tag {4.161}
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$$
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This condition in essence states that the deviatoric strain energy is to be bounded from below, a condition that is clearly satisfied. We further refer to and explain the ellipticity condition for the incompressible elasticity problem in Section 4.5.2.
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Let us emphasize that in this finite element formulation the only variables are the displacements.
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# Obtaining the Inf-Sup Condition
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The inf-sup condition—which when satisfied ensures that (4.156) holds—can now be developed as follows. Since the condition of total incompressibility clearly represents the most severe constraint, we consider this case. Then q = 0, u belongs to $K(q)$ for q = 0 [that is, $K(0)$ ], and the continuous problem (4.151) becomes
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$$
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\min _ {\mathbf {v} \in K (0)} \left\{\frac {1}{2} a (\mathbf {v}, \mathbf {v}) - \int_ {\mathrm{Vol}} \mathbf {f} ^ {B} \cdot \mathbf {v} d \mathrm{Vol} \right\} \tag {4.162}
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$$
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with the solution u, while the discrete problem is
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$$
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\min _ {\mathbf {v} _ {h} \in K _ {h} (0)} \left\{\frac {1}{2} a (\mathbf {v} _ {h}, \mathbf {v} _ {h}) - \int_ {\mathrm{Vol}} \mathbf {f} ^ {B} \cdot \mathbf {v} _ {h} d \mathrm{Vol} \right\} \tag {4.163}
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$$
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with the solution $\mathbf{u}_h$ .
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Now consider condition (4.156). We notice that in this condition we compare distances. In the following discussion we characterize a distance as “small” if it remains of the same order of magnitude as $d(\mathbf{u}, V_{h})$ as h decreases. Similarly, we will say that a vector is small if its length satisfies this definition and that a vector is “close” to another vector if the vector difference in the two vectors is small.
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Since $\mathbf{u}_h \in K_h(0)$ , and therefore always $\| \mathbf{u} - \mathbf{u}_h \| \leq \tilde{c} d[\mathbf{u}, K_h(0)]$ (see Exercise 4.47), we can also write condition (4.156) in the form
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$$
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d [ \mathbf {u}, K _ {h} (0) ] \leq c d (\mathbf {u}, V _ {h}) \tag {4.164}
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$$
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which means that we want the distance from $\mathbf{u}$ to $K_{h}(0)$ to be small. This relation expresses the requirement that if the distance between $\mathbf{u}$ and $V_{h}$ (the complete finite element displacement space) decreases at a certain rate as $h \to 0$ , then the distance between $\mathbf{u}$ and the space in which the actual solution lies [because $\mathbf{u}_{h} \in K_{h}(0)$ ] decreases at the same rate.
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Figure 4.23 shows schematically the spaces and vectors that we use. Let $u_{h0}$ be a vector of our choice in $K_{h}(0)$ and let $w_{h}$ be the corresponding vector such that
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$$
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\tilde {\mathbf {u}} _ {h} = \mathbf {u} _ {h 0} + \mathbf {w} _ {h} \tag {4.165}
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$$
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<details>
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<summary>text_image</summary>
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d[u, K_h(0)]
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d(u, V_h)
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u
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u_h0
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w_h
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\tilde{u}_h
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K_h(0)
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V_h
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</details>
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Figure 4.23 Spaces and vectors considered in deriving the inf-sup condition
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We can then prove that the condition in $(4.164)$ is fulfilled provided that
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for all $q_{h} \in D_{h}$ , there is a $\mathbf{w}_h \in K_h(q_h)$ such that
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$$
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\left\| \mathbf {w} _ {h} \right\| \leq c ^ {\prime} \left\| q _ {h} \right\| \tag {4.166}
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$$
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where $c'$ is independent of $h$ and the bulk modulus $\kappa$ .
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First, we always have (see Exercise 4.48)
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$$
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\| \operatorname{div} (\mathbf {u} - \tilde {\mathbf {u}} _ {h}) \| \leq \alpha \| \mathbf {u} - \tilde {\mathbf {u}} _ {h} \| \tag {4.167}
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$$
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and hence, $\| \operatorname{div} \tilde{\mathbf{u}}_h \| \leq \alpha d(\mathbf{u}, V_h)$ (4.168)
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where $\alpha$ is a constant and we used $\operatorname{div} \mathbf{u} = 0$ .
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Second, we consider
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$$
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\| \mathbf {u} - \mathbf {u} _ {h 0} \| = \| \mathbf {u} - \tilde {\mathbf {u}} _ {h} + \mathbf {w} _ {h} \|
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$$
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$$
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\leq \left\| \mathbf {u} - \tilde {\mathbf {u}} _ {h} \right\| + \left\| \mathbf {w} _ {h} \right\|
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$$
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Now assume that (4.166) holds with $q_{h} = div \tilde{u}_{h}$ . Because $div u_{h0} = 0$ , we have $div \tilde{u}_{h} = div w_{h}$ , where we note that $\tilde{u}_{h}$ is fixed by (4.155) and therefore $q_{h}$ is fixed, but by choosing different values of $u_{h0}$ different values of $w_{h}$ are also obtained. Then it follows that
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$$
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\begin{array}{l} \| \mathbf {u} - \mathbf {u} _ {h 0} \| \leq d (\mathbf {u}, V _ {h}) + c ^ {\prime} \| q _ {h} \| \\ = d (\mathbf {u}, V _ {h}) + c ^ {\prime} \| \operatorname{div} \tilde {\mathbf {u}} _ {h} \| \tag {4.169} \\ \leq d (\mathbf {u}, V _ {h}) + c ^ {\prime} \alpha d (\mathbf {u}, V _ {h}) \\ \end{array}
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$$
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We emphasize that we have used the condition (4.166) in this derivation and have assumed that $\mathbf{u}_{h0}$ is an element in $K_h(0)$ such that $\mathbf{w}_h$ satisfies (4.166). Also, note that (4.168) established only that $\| \operatorname{div} \tilde{\mathbf{u}}_h \|$ is small, but then (4.169) established that $\| \mathbf{u} - \mathbf{u}_{h0} \|$ is small.
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Third, since $\mathbf{u}_{h0} \in K_h(0)$ , we obtain from (4.169),
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$$
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d [ \mathbf {u}, K _ {h} (0) ] \leq \| \mathbf {u} - \mathbf {u} _ {h 0} \| \leq (1 + \alpha c ^ {\prime}) d (\mathbf {u}, V _ {h}) \tag {4.170}
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$$
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which is (4.164) with $c = 1 + \alpha c'$ , and we note that $c$ is independent of $h$ and the bulk modulus.
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The crucial step in the derivation of (4.164) is that using (4.166) with $q_{h} = \operatorname{div} \bar{u}_{h}$ , we can choose a vector $w_{h}$ that is small [which follows by using (4.166) and (4.168)]. We note that (4.166) is the only condition we need in order to prove (4.164) and is therefore the fundamental requirement to be satisfied in order to have a finite element discretization that will give an optimal rate of convergence.
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The optimal rate of convergence requires in (4.164) that the constant $c'$ in (4.166) be independent of h. Assume, for example, that instead of (4.166) we have $\|\mathbf{w}_{h}\|\leq(1/\beta_{h})\|q_{h}\|$ with $\beta_{h}$ decreasing with h. Then (4.170) will read
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$$
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d [ \mathbf {u}, K _ {h} (0) ] \leq \left(1 + \frac {\alpha}{\beta_ {h}}\right) d (\mathbf {u}, V _ {h}) \tag {4.171}
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$$
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and hence the distance between u and $K_{h}(0)$ will not decrease at the same rate as $d(\mathbf{u}, V_{h})$ . However, convergence, although not optimal, will still occur if $d(\mathbf{u}, V_{h})$ decreases faster than $\beta_{h}$ . This shows that the condition in (4.166) is a strong guarantee for good convergence properties of our discretization.
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Let us now rewrite (4.166) in the form of the inf-sup condition. From (4.166) we obtain, with $q_{h}$ and $w_{h}$ variables, $\mathbf{w}_{h} \in K_{h}(q_{h})$ , the condition
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$$
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\left\| \mathbf {w} _ {h} \right\| \left\| q _ {h} \right\| \leq c ^ {\prime} \left\| q _ {h} \right\| ^ {2} = c ^ {\prime} \int_ {\mathrm{Vol}} q _ {h} \operatorname{div} \mathbf {w} _ {h} d \mathrm{Vol} \tag {4.172}
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$$
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or the condition is that for all $q_{h} \in D_{h}$ , there is a $\mathbf{w}_h \in K_h(q_h)$ such that
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$$
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\frac {1}{c ^ {\prime}} \left\| q _ {h} \right\| \leq \frac {\int_ {\mathrm{Vol}} q _ {h} \operatorname{div} \mathbf {w} _ {h} d \mathrm{Vol}}{\left\| \mathbf {w} _ {h} \right\|} \tag {4.173}
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$$
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Hence, we want $\frac{1}{c'} \| q_h \| \leq \sup_{\mathbf{v}_h \in V_h} \frac{f_{\mathrm{Vol}} q_h \operatorname{div} \mathbf{v}_h d\mathrm{Vol}}{\|\mathbf{v}_h\|}$ (4.174)
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and the inf-sup condition follows,
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$$
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\inf _ {q _ {h} \in D _ {h}} \sup _ {\mathbf {v} _ {h} \in V _ {h}} \frac {\int_ {\mathrm{Vol}} q _ {h} \operatorname{div} \mathbf {v} _ {h} d \mathrm{Vol}}{\| \mathbf {v} _ {h} \| \| q _ {h} \|} \geq \beta > 0 \tag {4.175}
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$$
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with $\beta$ a constant independent of $\kappa$ and $h$
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We note that $\beta = 1 / c'$ .
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Therefore, (4.166) implies (4.175), and it can also be proven that (4.175) implies (4.166) (see Example 4.42). (We will not present this proof until later because we must first discuss certain additional basic facts.) Hence, we may also refer to (4.166) as one form of the inf-sup condition.
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The inf-sup condition says that for a finite element discretization to be effective, we must have that, for a sequence of finite element spaces, if we take any $q_{h} \in D_{h}$ , there must be a $v_{h} \in V_{h}$ such that the quotient in (4.175) is $\geq \beta > 0$ . If the inf-sup condition is satisfied by the sequence of finite element spaces, then our finite element discretization scheme will exhibit the good approximation property that we seek, namely, (4.156) will be fulfilled.
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Note that if $\beta$ is dependent on h, say (4.175) is satisfied with $\beta_{h}$ instead of $\beta$ , then the expression in (4.171) will be applicable (for an example, see the three-node isoparametric beam element in Section 4.5.7).
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Whether the inf-sup condition is satisfied depends, in general, on the specific finite element we use, the mesh topology, and the boundary conditions. If a discretization using a specific finite element always satisfies (4.175), for any mesh topology and boundary conditions, we simply say that the element satisfies the inf-sup condition. If, on the other hand, we know of one mesh topology and/or one set of (physically realistic) boundary conditions for which the discretization does not satisfy (4.175), then we simply say that the element does not satisfy the inf-sup condition.
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# Another Form of the Inf-Sup Condition
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To analyze whether an element satisfies the inf-sup condition (4.175), another form of this condition is very useful, namely
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For all $\mathbf{u}$ there is a $\mathbf{u}_l \in V_h$ (a vector that interpolates $\mathbf{u}$ ) such that
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$$
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\int_ {\mathrm{Vol}} \operatorname{div} (\mathbf {u} - \mathbf {u} _ {I}) q _ {h} d \mathrm{Vol} = 0 \quad \text { for all } q _ {h} \in D _ {h} \tag {4.176}
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$$
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$$
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\| \mathbf {u} _ {l} \| \leq c \| \mathbf {u} \|
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$$
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with the constant c independent of u, $u_{i}$ , and h.
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The equivalence of (4.176) and (4.175) [and hence (4.166)] can be formally proven (see F. Brezzi and M. Fortin [A] and F. Brezzi and K. J. Bathe [A, B]), but to simply relate the statements in (4.176) to our earlier discussion, we note that two fundamental requirements emerged in the derivation of the inf-sup condition; namely, that there is a vector $w_{h}$ such that (see Figure 4.23)
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$$
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\operatorname{div} \mathbf {w} _ {h} = \operatorname{div} \tilde {\mathbf {u}} _ {h} \tag {4.177}
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$$
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and [see (4.166) and (4.168)]
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$$
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\| \mathbf {w} _ {h} \| \leq c ^ {*} d (\mathbf {u}, V _ {h}) \tag {4.178}
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$$
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where $c^*$ is a constant.
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We note that (4.176) corresponds to (4.177) and (4.178) if we consider the vector $\tilde{u}_{h}-u$ (the vector of difference between the best approximation in $V_{h}$ and the exact solution) the solution vector and the vector $w_{h}$ the interpolation vector.
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Hence, the conditions are that the interpolation vector $w_{h}$ shall satisfy the above divergence and “small-size” conditions for and measured on the vector $(\tilde{\mathbf{u}}_{h}-\mathbf{u})$ in order to have an effective discretization scheme.
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The three expressions of the inf-sup condition, (4.166), (4.175), and (4.176), are useful in different ways but of course all express the same requirement. In mathematical analyses the forms (4.166) and (4.175) are usually employed, whereas (4.176) is frequently most easily used to prove whether a specific element satisfies the condition (see Example 4.36).
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Considering the inf-sup condition, we recognize that the richer the space $K_{h}(0)$ , the greater the capacity to satisfy (4.175) [that is, (4.164)]. However, unfortunately, using the standard displacement-based elements, the constraint is generally too strong for the elements and meshes (i.e., spaces $V_{h}$ ) of interest and the discretizations lock (see Fig. 4.20). We therefore turn to mixed formulations that do not lock and that exhibit the desired rates of convergence. Excellent candidates are the displacement/pressure formulations already introduced in Section 4.4.3. However, whereas the pure displacement formulation is (always) stable but generally locks, for any mixed formulation, a main additional concern is that it be stable. We shall see in the following discussion that the two conditions of stability and no locking are fulfilled if by appropriate choice of the displacement and pressure interpolations the ellipticity and inf-sup conditions are satisfied, and the desired (optimal) convergence rate is also obtained if the interpolations are chosen appropriately.
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# Weakening the Constraint
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Let us consider the $u / p$ formulation. The variational discrete problem in the $u / p$ formulation [corresponding to (4.140) and (4.143)] is
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$$
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\min _ {\mathbf {v} _ {h} \in \mathbf {V} _ {h}} \left\{\frac {1}{2} a \left(\mathbf {v} _ {h}, \mathbf {v} _ {h}\right) + \frac {\kappa}{2} \int_ {\text { Vol }} \left[ P _ {h} (\operatorname{div} \mathbf {v} _ {h}) \right] ^ {2} d \text { Vol } - \int_ {\text { Vol }} \mathbf {f} ^ {B} \cdot \mathbf {v} _ {h} d \text { Vol } \right\} \tag {4.179}
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$$
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where the projection operator $P_{h}$ is defined by
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$$
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\int_ {\mathrm{Vol}} \left[ P _ {h} (\operatorname{div} \mathbf {v} _ {h}) - \operatorname{div} \mathbf {v} _ {h} \right] q _ {h} d \mathrm{Vol} = 0 \quad \text { for all } q _ {h} \in Q _ {h} \tag {4.180}
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$$
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and $Q_{h}$ is a “pressure space” to be chosen. We see that $Q_{h}$ always contains $P_{h}(D_{h})$ but that $Q_{h}$ is sometimes larger than $P_{h}(D_{h})$ , which is a case that we shall discuss later.
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To recognize that (4.179) and (4.180) are indeed equivalent to the $u / p$ formulation, we rewrite (4.179) and (4.180) as
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$$
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2 G \int_ {\mathrm{Vol}} \epsilon_ {i j} ^ {\prime} (\mathbf {u} _ {h}) \epsilon_ {i j} ^ {\prime} (\mathbf {v} _ {h}) d \mathrm{Vol} - \int_ {\mathrm{Vol}} p _ {h} \operatorname{div} \mathbf {v} _ {h} d \mathrm{Vol} = \int_ {\mathrm{Vol}} \mathbf {f} ^ {B} \cdot \mathbf {v} _ {h} d \mathrm{Vol} \quad \forall \mathbf {v} _ {h} \in V _ {h} \tag {4.181}
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$$
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$$
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\int_ {\mathrm{Vol}} \left(\frac {p _ {h}}{\kappa} + \operatorname{div} \mathbf {u} _ {h}\right) q _ {h} d \mathrm{Vol} = 0 \quad \forall q _ {h} \in Q _ {h} \tag {4.182}
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$$
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These equations are (4.140) and (4.143) in Section 4.4.3, and we recall that they are valid for any value of $\kappa > 0$ . The key point in the u/p formulation is that (4.180) [i.e., (4.182)] is applied individually for each element and, provided $\kappa$ is finite, the pressure variables can be statically condensed out on the element level (before assembly of the element stiffness matrix into the global structure stiffness matrix).
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Consider the following example.
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EXAMPLE 4.34: Derive $P_{h}(\text{div } \mathbf{v}_{h})$ for the 4/1 element shown in Fig. E4.34. Hence, evaluate the term $(\kappa/2) \int_{\text{vol}} [P_{h}(\text{div } \mathbf{v}_{h})]^{2} \, d\text{Vol}$ in (4.179).
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<details>
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<summary>text_image</summary>
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</details>
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Figure E4.34 A 4/1 plane strain element
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We have
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$$
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\operatorname{div} \mathbf {v} _ {h} = \left[ h _ {1, x} \quad \dots \quad h _ {4, x} \quad \vdots \quad h _ {1, y} \quad \dots \quad h _ {4, y} \right] \hat {\mathbf {u}}
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$$
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where $\hat{\mathbf{u}}^T = [u_1\ldots u_4\vdots v_1\ldots v_4]$
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We now use (4.180), with $q_{h}$ an arbitrary nonzero constant (say $q_{h} = \alpha$ ), because here $Q_{h}$ is the space of constant pressures. Since $P_{h}(\text{div } \mathbf{v}_{h})$ is also constant, we have from (4.180),
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$$
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4 P _ {h} (\operatorname{div} \mathbf {v} _ {h}) \alpha = \alpha \int_ {\mathrm{vol}} \operatorname{div} \mathbf {v} _ {h} d \mathrm{Vol}
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$$
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which gives $P_{h}(\mathrm{div}\mathbf{v}_{h}) = \frac{1}{4} [1 - 1 - 1 1:1 1 - 1 - 1]\hat{\mathbf{u}}$
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$$
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= \mathbf {D} \hat {\mathbf {u}}
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$$
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Hence, $\frac{\kappa}{2}\int_{\mathrm{Vol}}\left[P_h(\mathrm{div}\mathbf{v}_h)\right]^2 d\mathrm{Vol} = \frac{\kappa}{2}\hat{\mathbf{u}}^T\mathbf{G}_h\hat{\mathbf{u}}$
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where $\mathbf{G}_h = 4\mathbf{D}^T\mathbf{D}$
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Note that although we have used the pressure space $Q_{h}$ , the stiffness matrix obtained from (4.179) will correspond to nodal point displacements only.
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Also, we may note that the term $P_{h}(\text{div } \mathbf{v}_{h})$ is simply div $v_{h}$ at x = y = 0.
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EXAMPLE 4.35: Consider the nine-node element shown in Fig. E4.35 and assume that $v_{h}$ is given by the nodal point displacements $u_{1}=1$ , $u_{5}=0.5$ , $u_{8}=0.5$ , $u_{9}=0.25$ with all other nodal point displacements equal to zero. Let $Q_{h}$ be the space corresponding to $\{1, x, y\}$ . Evaluate $P_{h}(\text{div } v_{h})$ .
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To evaluate $P_{h}(\text{div } \mathbf{v}_{h})$ we use the general relationship
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$$
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\int_ {\mathrm{Vol}} (P _ {h} (\mathrm{div} \mathbf {v} _ {h}) - \mathrm{div} \mathbf {v} _ {h}) q _ {h} d \mathrm{Vol} = 0 \qquad \forall q _ {h} \in Q _ {h} \tag {a}
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$$
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In this example,
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$$
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\operatorname{div} \mathbf {v} _ {h} = \frac {\partial u _ {h}}{\partial x} + \frac {\partial v _ {h}}{\partial y}
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$$
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<details>
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<summary>text_image</summary>
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Figure E4.35 A 9/3 element subjected to nodal point displacements
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where $u_{h}$ and $v_{h}$ are given by the element nodal point displacements. Hence,
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$$
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u _ {h} = \frac {1}{4} (1 + x) (1 + y)
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$$
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$$
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v _ {h} = 0
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$$
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and $\operatorname{div} \mathbf{v}_h = \frac{1}{4}(1 + y)$
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Let $P_{h}(\mathrm{div}\mathbf{v}_{h}) = a_{1} + a_{2}x + a_{3}y$
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then (a) gives $\int_{\mathrm{Vol}}\left[(a_1 + a_2x + a_3y) - \frac{1}{4} (1 + y)\right]q_h dx dy = 0$ (b)
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for $q_{h} = 1, x$ , and $y$ . Hence, (b) gives the set of equations
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$$
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\left[ \begin{array}{c c c} \int_ {\text {Vol}} d x d y & \int_ {\text {Vol}} x d x d y & \int_ {\text {Vol}} y d x d y \\ & \int_ {\text {Vol}} x ^ {2} d x d y & \int_ {\text {Vol}} x y d x d y \\ \text {Symmetric} & & \int_ {\text {Vol}} y ^ {2} d x d y \end{array} \right] \left[ \begin{array}{l} a _ {1} \\ a _ {2} \\ a _ {3} \end{array} \right] = \left[ \begin{array}{l} \int_ {\text {Vol}} \frac {1}{4} (1 + y) d x d y \\ \int_ {\text {Vol}} \frac {1}{4} (1 + y) x d x d y \\ \int_ {\text {Vol}} \frac {1}{4} (1 + y) y d x d y \end{array} \right]
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$$
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or $\left[ \begin{array}{ccc}4 & 0 & 0\\ & \frac{4}{3} & 0\\ \text{Sym.} & & \frac{4}{3} \end{array} \right]\left[ \begin{array}{c}a_{1}\\ a_{2}\\ a_{3} \end{array} \right] = \left[ \begin{array}{c}1\\ 0\\ \frac{1}{3} \end{array} \right]$ (c)
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The solution of (c) gives $a_1 = \frac{1}{4}$ , $a_2 = 0$ , $a_3 = \frac{1}{4}$ , and hence,
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$$
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P _ {h} (\operatorname{div} \mathbf {v} _ {h}) = \frac {1}{4} (1 + y)
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$$
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This result is correct because $\operatorname{div} \mathbf{v}_h$ can be represented exactly in $Q_h$ and in such a case the projection gives of course the value of $\operatorname{div} \mathbf{v}_h$ .
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The inf-sup condition corresponding to (4.179) is now like the inf-sup condition we discussed earlier but using the term $P_{h}(\text{div } \mathbf{v}_{h})$ instead of div $v_{h}$ . Hence our condition is now
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$$
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\inf _ {q _ {h} \in P _ {h} (D _ {h})} \sup _ {\mathbf {v} _ {h} \in V _ {h}} \frac {\int_ {\mathrm{Vol}} q _ {h} \operatorname{div} \mathbf {v} _ {h} d \mathrm{Vol}}{\| \mathbf {v} _ {h} \| \| q _ {h} \|} \geq \beta > 0 \tag {4.183}
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$$
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In other words, the inf-sup condition now corresponds to any element in $V_{h}$ and $P_{h}(D_{h})$ . Hence, when applying (4.166), (4.175), or (4.176) to the mixed interpolated u/p elements, we now need to consider the finite element spaces $V_{h}$ and $P_{h}(D_{h})$ , where $P_{h}(D_{h})$ is used instead of $D_{h}$ .
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EXAMPLE 4.36: Prove that the inf-sup condition is satisfied by the 9/3 two-dimensional $u / p$ element presented in Section 4.4.3.
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For this proof we use the form of the inf-sup condition given in (4.176) (see F. Brezzi and K. J. Bathe [A]). Given u smooth we must find an interpolation, $u_{I} \in V_{h}$ , such that for each element m,
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$$
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\int_ {\mathrm{Vol} ^ {(m)}} (\operatorname{div} \mathbf {u} - \operatorname{div} \mathbf {u} _ {l}) q _ {h} d \mathrm{Vol} ^ {(m)} = 0 \tag {a}
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$$
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for all $q_{h}$ polynomials of degree $\leq1$ in $\mathrm{Vol}^{(m)}$ . To define $u_{l}$ we prescribe the values of each displacement at the nine element nodes (corner nodes, midside nodes, and the center node). We start with the corner nodes and require for these nodes i=1,2,3,4,
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$$
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\left. \mathbf {u} _ {I} \right| _ {i} = \left. \mathbf {u} \right| _ {i} \quad \text { eight conditions } \tag {b}
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$$
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Then we adjust the values at the midside nodes $j = 5, 6, 7, 8$ in such a way that
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$$
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\int_ {s _ {j}} \left(\mathbf {u} - \mathbf {u} _ {I}\right) \cdot \mathbf {n} d S = \int_ {s _ {j}} \left(\mathbf {u} - \mathbf {u} _ {I}\right) \cdot \boldsymbol {\tau} d S = 0 \quad \text { eight conditions } \tag {c}
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$$
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for every edge $S_1, \ldots, S_4$ of the element with $\mathbf{n}$ the unit normal vector and $\tau$ the unit tangential vector to the edge.
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Next we note that (a) in particular implies, for every constant $q_{h}$ ,
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$$
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\int_ {\mathrm{Vol} ^ {(m)}} \operatorname{div} (\mathbf {u} - \mathbf {u} _ {I}) q _ {h} d \mathrm{Vol} ^ {(m)} = q _ {h} \sum_ {s _ {1}, \dots , s _ {4}} \int_ {s _ {j}} (\mathbf {u} - \mathbf {u} _ {I}) \cdot \mathbf {n} d S \tag {d}
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$$
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We are left to use the two degrees of freedom at the element center node. We choose these in such a way that
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$$
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\int_ {\operatorname{Vol} ^ {(m)}} \operatorname{div} (\mathbf {u} - \mathbf {u} _ {I}) x d \operatorname{Vol} ^ {(m)} = \int_ {\operatorname{Vol} ^ {(m)}} \operatorname{div} (\mathbf {u} - \mathbf {u} _ {I}) y d \operatorname{Vol} ^ {(m)} = 0 \tag {e}
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$$
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We note now that (d) and (e) imply (a) and that $\mathbf{u}_I$ , constructed element by element through (b) and (c), will be continuous from element to element. Finally, note that clearly if $\mathbf{u}$ is a (vector) polynomial of degree $\leq 2$ on the element, we obtain $\mathbf{u}_I \equiv \mathbf{u}$ and this ensures optimal bounds for $\| \mathbf{u} - \mathbf{u}_I \|$ and implies the condition $\| \mathbf{u}_I \| \leq c \| \mathbf{u} \|$ in (4.176) for all $\mathbf{u}$ .
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While in the u/p formulation the projection (4.180) is carried out for each element individually, in the u/p-c formulation a continuous pressure interpolation is assumed and then (4.181) and (4.182) are applied. The relation (4.182) with the continuous pressure interpolation gives a set of equations coupling the displacements and pressures for adjacent
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elements. In this case the inf-sup condition is still given by (4.183), but now the pressure space corresponds to the nodal point continuous pressure interpolations.
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In dealing with the inf-sup condition, we recognize that the ability to satisfy the condition depends on how the space $P_{h}(D_{h})$ relates to the space of displacements $V_{h}$ . Here again, $P_{h}$ is the projection operator onto the space $Q_{h}$ [see (4.180) and (4.182)], and, in general, the smaller the space $Q_{h}$ , the easier it is to satisfy the condition. Of course, if for a given space $V_{h}$ the inf-sup condition is satisfied with $Q_{h}$ smaller than necessary, we have a stable element but the predictive capability is not as high as possible (namely, as high as it would be using the larger space $Q_{h}$ but still satisfying the inf-sup condition).
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For example, consider the nine-node isoparametric element (see Section 4.4.3). Using the u/p formulation with $P_{h} = I$ (the identity operator), the displacement-based formulation is obtained and the element locks. Reducing the constraint to obtain the 9/3 element, the inf-sup condition is satisfied (see Example 4.36) and optimal convergence rates are obtained for the displacements and the pressure; that is, the convergence rate for the displacements is $o(h^{3})$ and for the stress is $o(h^{2})$ , which is all that we can expect with a parabolic interpolation of displacements and a linear interpolation of pressure. Reducing the constraint further to obtain the 9/1 element, the inf-sup condition is also satisfied, and while the element behavior for the interpolations used is still optimal, the predictive capability of this nine-node element is not the best possible (because a constant element pressure is assumed, whereas a linear pressure variation could be used).
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This observation (about the quality of the solution) is explained by the error bounds (see, for example, F. Brezzi and K. J. Bathe [B]). Let $u_{I} \in V_{h}$ be an interpolant of u satisfying
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$$
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\left. \begin{array}{l} \int_ {\mathrm{Vol}} \left[ \operatorname{div} \left(\mathbf {u} - \mathbf {u} _ {I}\right) \right] q _ {h} d \mathrm{Vol} = 0 \quad \forall q _ {h} \in P _ {h} \left(D _ {h}\right) \\ \| \mathbf {u} _ {i} \| \leqslant c \| \mathbf {u} \| \end{array} \right\} \tag {4.184}
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$$
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and $\| \mathbf{u}_I\| \leq c\| \mathbf{u}\|$
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If (4.184) holds for all possible solutions u, then
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$$
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\| \mathbf {u} - \mathbf {u} _ {h} \| \leq c _ {1} (\| \mathbf {u} - \mathbf {u} _ {l} \| + \| (I - P _ {h}) p \|) \tag {4.185}
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$$
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and $\| p + \kappa P_h(\mathrm{div} \mathbf{u}_h) \| \leq c_2(\|\mathbf{u} - \mathbf{u}_I\| + \| (I - P_h)p\|)$ (4.186)
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where $p = -\kappa$ div $\mathbf{u}$ and $c_1, c_2$ are constants independent of $h$ and $\kappa$ . We note of course that (4.184) is the inf-sup condition with the weakened constraint $q_h \in P_h(D_h)$ [see (4.176)] and that the right-hand sides of (4.185) and (4.186) are smaller the closer $P_h$ is to $I$ .
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# 4.5.2 The Inf-Sup Condition Derived from the Matrix Equations
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Further insight into the inf-sup condition is obtained by studying the governing algebraic finite element equations. Let us consider the case of total incompressibility (it being the most severe case),
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$$
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\left[ \begin{array}{c c} \left(\mathbf {K} _ {u u}\right) _ {h} & \left(\mathbf {K} _ {u p}\right) _ {h} \\ \left(\mathbf {K} _ {p u}\right) _ {h} & \mathbf {0} \end{array} \right] \left[ \begin{array}{l} \mathbf {U} _ {h} \\ \mathbf {P} _ {h} \end{array} \right] = \left[ \begin{array}{c} \mathbf {R} _ {h} \\ \mathbf {0} \end{array} \right] \tag {4.187}
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$$
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where $U_{h}$ lists all the unknown nodal point displacements and $P_{h}$ lists the unknown pressure variables. Since the material is assumed to be totally incompressible, we have a square null
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matrix equal in size to the number of pressure variables in the lower right of the coefficient matrix.
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The mathematical analysis of the formulation resulting in (4.187) consists of a study of the solvability and the stability of the equations, where the stability of the equations implies their solvability.
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The solvability of (4.187) simply refers to the fact that (4.187) can actually be solved for unique vectors $\mathbf{U}_h$ and $\mathbf{P}_h$ when $\mathbf{R}_h$ is given.
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The conditions for solvability (see Exercise 4.54) are
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Condition i:
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$$
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\mathbf {V} _ {h} ^ {T} (\mathbf {K} _ {u u}) _ {h} \mathbf {V} _ {h} > 0 \quad \text { for all } \mathbf {V} _ {h} \text { satisfying } (\mathbf {K} _ {p u}) _ {h} \mathbf {V} _ {h} = \mathbf {0} \tag {4.188}
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$$
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Condition ii:
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$$
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(\mathbf {K} _ {u p}) _ {h} \mathbf {Q} _ {h} = \mathbf {0} \quad \text { implies that } \mathbf {Q} _ {h} \text { must be zero } \tag {4.189}
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$$
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The space of displacement vectors $V_{h}$ that satisfy $(\mathbf{K}_{pu})_{h}\mathbf{V}_{h}=0$ represents the kernel of $(\mathbf{K}_{pu})_{h}$ .
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The stability of the formulation is studied by considering a sequence of problems of the form (4.187) with increasingly finer meshes. Let S be the smallest constant such that
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$$
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\frac {\left\| \Delta \mathbf {u} _ {h} \right\| _ {V} + \left\| \Delta p _ {h} \right\| _ {0}}{\left\| \mathbf {u} _ {h} \right\| _ {V} + \left\| p _ {h} \right\| _ {0}} \leq S \frac {\left\| \Delta \mathbf {f} ^ {B} \right\| _ {D V}}{\left\| \mathbf {f} ^ {B} \right\| _ {D V}} \tag {4.190}
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$$
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for all $\mathbf{u}_h, p_h, \mathbf{f}^B, \Delta \mathbf{u}_h, \Delta p_h, \Delta \mathbf{f}^B$ , where $\| \cdot \|_V$ and $\| \cdot \|_0$ are the norms defined in (4.153), $\| \cdot \|_{DV}$ means the dual norm of $\| \cdot \|_V$ (see Section 2.7), and $\Delta \mathbf{f}^B, \Delta \mathbf{u}_h$ , and $\Delta p_h$ denote a prescribed perturbation on the load function $\mathbf{f}^B$ and the resulting perturbations on the displacement vector $\mathbf{u}_h$ and pressure $p_h$ . Of course, we have
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$$
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\left[ \begin{array}{c c} \left(\mathbf {K} _ {u u}\right) _ {h} & \left(\mathbf {K} _ {u p}\right) _ {h} \\ \left(\mathbf {K} _ {p u}\right) _ {h} & \mathbf {0} \end{array} \right] \left[ \begin{array}{l} \Delta \mathbf {U} _ {h} \\ \Delta \mathbf {P} _ {h} \end{array} \right] = \left[ \begin{array}{c} \Delta \mathbf {R} _ {h} \\ \mathbf {0} \end{array} \right] \tag {4.191}
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$$
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where $\Delta R_{h}$ corresponds to the load variation $\Delta f^{B}$ and the norms of the finite element variables in (4.190) are given by the nodal point values listed in the solution vectors. Hence (4.190) expresses that for a given relative perturbation in the load vector, the corresponding relative perturbation in the solution is bounded by S times the relative perturbation in the loads.
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For any given fixed mesh, satisfying the conditions of solvability (4.188) and (4.189) implies that (4.190) is satisfied for some S, the value of which depends on the mesh.
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The formulation is stable if for any sequence of meshes the stability constant S is uniformly bounded. Hence, our question of stability reduces to asking for the conditions on the matrices $(\mathbf{K}_{uu})_{h}$ and $(\mathbf{K}_{up})_{h}$ that ensure that S remains uniformly bounded when using any sequence of meshes.
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We considered briefly in Section 2.7 the stability conditions as related to a formulation that leads to a general coefficient matrix A [see (2.169) to (2.179)]. If we specialize these considerations to the specific coefficient matrix used in the displacement/pressure formulations, we will find a rather natural result (see F. Brezzi and K. J. Bathe [B]), namely, that the stability conditions are an extension of the solvability conditions (4.188) and (4.189) in that stability in the use of these relations with increasingly finer meshes must be preserved.
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