김경종
33 KiB
The stability condition corresponding to the solvability condition (4.188) is that there is an \alpha > 0 independent of the mesh size such that
\mathbf {V} _ {h} ^ {T} (\mathbf {K} _ {u u}) _ {h} \mathbf {V} _ {h} \geq \alpha \| \mathbf {v} _ {h} \| _ {V} ^ {2} \quad \text { for all } \mathbf {V} _ {h} \in \text { kernel } \left[ (\mathbf {K} _ {p u}) _ {h} \right] \tag {4.192}
This condition is the ellipticity condition already mentioned briefly in Section 4.5.1. The relation states that, for any fineness of mesh, the Rayleigh quotient obtained with any vector V_{h} satisfying (\mathbf{K}_{pu})_{h}\mathbf{V}_{h} = \mathbf{0} , will be bounded from below by the constant \alpha (which is independent of element mesh size). This ellipticity condition is quite easily fulfilled (by the choice of a high enough pressure interpolation) in our displacement/pressure formulations. We elaborate upon this fact in the following example.
EXAMPLE 4.37: Consider the ellipticity condition in (4.192). Discuss that it can be satisfied for any (practical) displacement/pressure formulation.
To understand that the ellipticity condition can be fulfilled, we need to recall that (4.187) is the result of the finite element discretization in (4.179). Hence,
\mathbf {V} _ {h} ^ {T} (\mathbf {K} _ {u u}) _ {h} \mathbf {V} _ {h}; \quad \mathbf {V} _ {h} \in \text { kernel } (\mathbf {K} _ {p u}) _ {h} \tag {a}
corresponds to twice the strain energy stored in the finite element discretization when v_{h} corresponds to an element in V_{h} that satisfies P_{h}(\text{div } \mathbf{v}_{h}) = 0 . Hence, by selecting the pressure space Q_{h} large enough the expression in (a) will always be greater than zero (and bounded from below). For example, for the 8/1 axisymmetric and 20/1 3-D elements in Table 4.6, the pressure space is not large enough.
If (4.192) is not satisfied, we could, also, stabilize the solution. This is achieved by considering the almost incompressible case and using the variational formulation
\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(\operatorname{div} \mathbf {v} _ {h}\right) ^ {2} d \text { Vol } + \frac {\kappa - \kappa^ {*}}{2} \int_ {\text { Vol }} \left[ P _ {h} \left(\operatorname{div} \mathbf {v} _ {h}\right) \right] ^ {2} d \text { Vol } - \int_ {\text { Vol }} \mathbf {f} ^ {\beta} \cdot \mathbf {v} _ {h} d \text { Vol } \right\}
where \kappa^{*} is a bulk modulus of the order of the shear modulus and does not lead to locking. Of course, we could now assume (\kappa - \kappa^{*}) \to \infty .
This procedure amounts to evaluating a portion of the bulk energy as in the displacement method and using a projection for the remaining portion. Note that when \kappa is equal to \kappa^{*} , the part to be projected is zero. Hence the essence of the scheme is that a well-behaved part of the term that is difficult to deal with has been moved to be evaluated without the projection. This kind of stabilization to satisfy the ellipticity condition can be important in the design of formulations (see F. Brezzi and M. Fortin [A]). The procedure has been proposed to stabilize a displacement/pressure formulation for the analysis of inviscid fluids (see C. Nitikitpaiboon and K. J. Bathe [A]) and for the development of plate and shell elements (see D. N. Arnold and F. Brezzi [A]). However, the difficulty with this approach can be in selecting the portions of energies to be evaluated with and without projection, in particular when the various kinematic actions are fully coupled as, for instance, in the analysis of shell structures (see Section 5.4.2).
The stability condition corresponding to the solvability condition (4.189) is that there is a \beta > 0 independent of the mesh size h such that
\inf _ {\mathbf {Q} _ {h}} \sup _ {\mathbf {v} _ {h}} \frac {\mathbf {Q} _ {h} ^ {T} (\mathbf {K} _ {p u}) _ {h} \mathbf {V} _ {h}}{\| q _ {h} \| \| \mathbf {v} _ {h} \|} \geq \beta > 0 \tag {4.193}
for every problem in the sequence.
Note that here we take the sup using the elements in V_{h} and the inf using the elements in Q_{h} . Of course, this relation is our inf-sup condition (4.183) in algebraic form, but we now have q_{h} \in Q_{h} , where Q_{h} is not necessarily equal to P_{h}(D_{h}) .
We note that a simple test consisting of counting displacement and pressure variables and comparing the number of such variables is not adequate to identify whether a mixed formulation is stable. The above discussion shows that such a test is certainly not sufficient to ensure the stability of a formulation and in general does not even ensure that condition (4.189) for solvability is satisfied (see also Exercises 4.60 and 4.64).
4.5.3 The Constant (Physical) Pressure Mode
Let us assume in this section that our finite element discretization contains no spurious pressure modes (which we discuss in the next section) and that the inf-sup condition for q_{h} \in P_{h}(D_{h}) is satisfied.
We mentioned earlier (see Section 4.4.3) that when our elasticity problem corresponds to total incompressibility (i.e., we consider q = \operatorname{div} u = 0 ) and all displacements normal to the surface of the body are prescribed (i.e., S_{u} is equal to S), special considerations are necessary. Actually, we can consider the following two cases.
Case i: All displacements normal to the body surface are prescribed to be zero. In this case, the pressure is undetermined unless it is prescribed at one point in the body. Namely, assume that p_{0} is a constant pressure. Then
\int_ {\mathrm{Vol}} p _ {0} \operatorname{div} \mathbf {v} _ {h} d \mathrm{Vol} = p _ {0} \int_ {S} \mathbf {v} _ {h} \cdot \mathbf {n} d S = 0 \quad \forall \mathbf {v} _ {h} \in V _ {h} \tag {4.194}
where n is the unit normal vector to the body surface. Hence, if the pressure is not prescribed at one point, we can add an arbitrary constant pressure p_{0} to any proposed solution. A consequence is that the equations (4.187) cannot be solved unless the pressure is prescribed at one point, which amounts to eliminating one pressure degree of freedom [one column in (\mathbf{K}_{up})_{h} and the corresponding row in (\mathbf{K}_{pu})_{h} ]. If this pressure degree of freedom is not eliminated, Q_{h} is larger than P_{h}(D_{h}) , the solvability condition (4.189) is not satisfied, and the inf-sup value including this pressure mode is zero. For a discussion of the case Q_{h} larger than P_{h}(D_{h}) but pertaining to spurious pressure modes, see Section 4.5.4.
Of course, instead of eliminating one pressure degree of freedom, it may be more expedient in practice to release some displacement degrees of freedom normal to the body surface.
Case ii: All displacements normal to the body surface are prescribed with some nonzero values. The difficulty in this case is that the incompressibility condition must be fulfilled
\int_ {\mathbf {v} \mathrm{ol}} \mathrm{div} \mathbf {v} _ {h} d \mathrm{Vol} = \int_ {S} \mathbf {v} _ {h} \cdot \mathbf {n} d S = 0 \quad \forall \mathbf {v} _ {h} \in V _ {h} \tag {4.195}
A constant pressure mode will also be present, which can be eliminated as discussed for Case i. If the body geometry is complex, it can be difficult to satisfy exactly the surface integral condition in (4.195). Since any error in fulfilling this condition can result in a large error in pressure prediction, it may be best in practice to leave the displacement(s) normal to the surface free at some node(s).
Let us next consider that the body is only almost incompressible, that \kappa is large but finite, and that the u/p formulation is used. In Case i, the arbitrary constant pressure p_{0} will then automatically be set to zero (in the same way as spurious modes are set to zero; see Section 4.5.4). This is a most convenient result because we do not need to be concerned with the elimination of a pressure degree of freedom. Of course, in practice we could also leave some nodal point displacement degree(s) of freedom normal to the body surface free, which would eliminate the constant pressure mode.
With the constant pressure mode present in the model, Q_{h} is (by one basis vector) larger than P_{h}(D_{h}) and the inf-sup value corresponding to this mode is zero. Nevertheless, we can solve the algebraic equations and obtain a reliable solution (unless \kappa is so large that the ill-conditioning of the coefficient matrix results in significant round-off errors, see Section 8.2.6).
In Case ii, it is probably best to proceed as recommended above, namely, to leave some nodal displacement(s) normal to the surface free, in order to give the material the freedom to satisfy the constraint of near incompressibility. Then the constant pressure mode is not present in the finite element model.
An important point in these considerations is that if all displacements normal to the surface of the body are prescribed, the pressure space will be larger than P_{h}(D_{h}) , but only by the constant pressure mode. This mode is of course a physical phenomenon and not a spurious mode. If the inf-sup condition for q_{h} \in P_{h}(D_{h}) is satisfied, then the solution is rendered stable and accurate by simply eliminating the constant pressure mode (or using the u/p formulation with a not too large value of \kappa to automatically set the value of the constant pressure to zero). We consider in the next section the case of Q_{h} larger than P_{h}(D_{h}) as a result of spurious pressure modes.
4.5.4 Spurious Pressure Modes—The Case of Total Incompressibility
We consider in this section the condition of total incompressibility and, merely for simplicity of discussion, that the physical constant pressure mode mentioned earlier is not present in the model. If it were actually present, the considerations given above would apply in addition to those we shall now present.
With this provision, we recall that in our discussion of the inf-sup condition we assumed that the space Q_{h} is equal to the space P_{h}(D_{h}) [see (4.183)], whereas in (4.193) we have no such restriction. In an actual finite element solution we may well have P_{h}(D_{h}) \subsetneq Q_{h} , and it is important to recognize the consequences.
If the space Q_{h} is larger than the space P_{h}(D_{h}) , the solution will exhibit spurious pressure modes. These modes are a result of the numerical solution procedure only, namely, the specific finite elements and mesh patterns used, and have no physical explanation.
We define a spurious pressure mode as a (nonzero) pressure distribution p_{s} that satisfies the relation
\int_ {\mathrm{Vol}} p _ {s} \operatorname{div} \mathbf {v} _ {h} d \mathrm{Vol} = 0 \quad \forall \mathbf {v} _ {h} \in V _ {h} \tag {4.196}
In the matrix formulation (4.187) a spurious pressure mode corresponds to the case
\left(\mathbf {K} _ {u p}\right) _ {h} \mathbf {P} _ {s} = \mathbf {0} \tag {4.197}
where P_{s} is the (nonzero) vector of pressure variables corresponding to p_{s} . Hence, the solvability condition (4.189) is not satisfied when spurious pressure modes are present, and of course the inf-sup value when testing over the complete space Q_{h} in (4.193) is zero.
Let us show that if Q_{h} is equal to P_{h}(D_{h}) , there can be no spurious pressure mode. Assume that \hat{p}_{h} is proposed to be a spurious pressure mode. If Q_{h} = P_{h}(D_{h}) , there is always a vector \hat{v}_{h} such that \hat{p}_{h} = -P_{h} (div \hat{v}_{h} ). However, using \hat{v}_{h} in (4.196), we obtain
- \int_ {\text {Vol}} \hat {p} _ {h} \operatorname{div} \hat {\mathbf {v}} _ {h} d \text {Vol} = - \int_ {\text {Vol}} \hat {p} _ {h} P _ {h} (\operatorname{div} \hat {\mathbf {v}} _ {h}) d \text {Vol} = \int_ {\text {Vol}} \hat {p} _ {h} ^ {2} d \text {Vol} > 0 \tag {4.198}
meaning that (4.196) is not satisfied. On the other hand, if Q_{h} is greater than P_{h}(D_{h}) , notably P_{h}(D_{h}) \subsetneq Q_{h} , then we can find a pressure distribution in the space orthogonal to P_{h}(D_{h}) , and hence for that pressure distribution (4.196) is satisfied (see Example 4.38).
Hence, we now recognize that in essence we have two phenomena that may occur when testing a specific finite element discretization using displacements and pressure as variables:
- The locking phenomenon, which is detected by the smallest value of the inf-sup expression not being bounded from below by a value
\beta > 0[see discussion following (4.156)] - The spurious modes phenomenon, which corresponds to a zero value of the inf-sup expression when we test with
q_{h} \in Q_{h}.
Of course, when a discretization with spurious modes is considered, we might still be interested in the smallest nonzero value of the inf-sup expression, and we can focus on this value by only testing with q_{h} \in P_{h}(D_{h}) , in other words, by ignoring all spurious pressure modes.
The numerical inf-sup test described in Section 4.5.6 actually gives the smallest nonzero value of the inf-sup expression and also evaluates the number of spurious pressure modes.
Let us note here, as a side remark, that the spurious pressure modes have no relationship to the spurious zero energy modes mentioned in Section 5.5.6 (and which are a result of using reduced or selective numerical integration in the evaluation of element stiffness matrices). In the displacement/pressure formulations considered here, each element stiffness matrix is accurately calculated and exhibits only the correct physical rigid body modes. The spurious pressure modes in the complete mesh are a result of the specific displacement and pressure spaces used for the complete discretization.
One way to gain more insight into the relation (4.193) is to imagine the matrix (\mathbf{K}_{up})_h [or (\mathbf{K}_{pu})_h = (\mathbf{K}_{up})_h^T] in diagonalized form (choosing the appropriate basis for displacements
and pressure variables), in which case we would have
text_image
(K_up)h = \n√λ_n\n√λ{n-1}^\dagger\n\sqrt{λ_k}\n0\n0\n0\nElements\nnot shown\nare zeros\nnp\nKernel (K_up)_h\nn_u\nKernel (K_pu)_h\n(4.199)
^{\dagger} We call the elements \sqrt{\lambda_{i}} in anticipation of our discussion in Section 4.5.6.
In this representation the zero columns define the kernel of (\mathbf{K}_{up})_h and each zero column corresponds to a spurious pressure mode. Also, since for any displacement vector \hat{\mathbf{U}}_h we need
(\mathbf {K} _ {p u}) _ {h} \hat {\mathbf {U}} _ {h} = \mathbf {0} \tag {4.200}
and (\mathbf{K}_{pu})_{h} = (\mathbf{K}_{up})_{h}^{T} , the size of the kernel of (\mathbf{K}_{pu})_{h} determines whether the solution is overconstrained. Whereas, on one hand, we want the kernel of (\mathbf{K}_{up})_{h} to be zero (no spurious pressure modes), on the other hand, we want the kernel of (\mathbf{K}_{pu})_{h} to be large so as to admit many linearly independent vectors \hat{U}_{h} that satisfy (4.200). Our actual displacement solution to the problem (4.187) will lie in the subspace spanned by these vectors, and if that subspace is too small, as a result of the pressure space Q_{h} being too large, the solution will be overconstrained. The theory on the inf-sup condition [see the discussion in Section 4.5.1 and (4.193)] showed that this overconstraint is detected by \sqrt{\lambda_{k}} decreasing to zero as the mesh is refined. Vice versa, if \sqrt{\lambda_{k}} \geq \beta > 0 , for any mesh, as the size of the elements is decreased, with \beta independent of the mesh, the solution space is not overconstrained and the discretization yields a reliable solution (with the optimal rate of convergence in the displacements and pressure, provided the pressure space is largest without violating the inf-sup condition; see Section 4.5.1).
4.5.5 Spurious Pressure Modes—The Case of Near Incompressibility
In the above discussion we assumed conditions of total incompressibility, and the use of either the u/p or the u/p-c formulation. Consider now that we have a finite (but large) \kappa and that the u/p formulation with static condensation on the pressure degrees of freedom (as is typical) is used. In this case, the governing finite element equations are, for a typical element
(or the complete mesh),
\left[ \begin{array}{l l} \left(\mathbf {K} _ {u u}\right) _ {h} & \left(\mathbf {K} _ {u p}\right) _ {h} \\ \left(\mathbf {K} _ {p u}\right) _ {h} & \left(\mathbf {K} _ {p p}\right) _ {h} \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 {4.201}
or \left[(\mathbf{K}_{uu})_h - (\mathbf{K}_{up})_h(\mathbf{K}_{pp})_h^{-1}(\mathbf{K}_{pu})_h\right]\mathbf{U}_h = \mathbf{R}_h (4.202)
So far we have assumed that no nonzero displacements are prescribed. It is an important observation that in this case any spurious pressure mode has no effect on the predicted displacements and pressure. The reason can be shown by considering (\mathbf{K}_{up})_{h} in (4.199) with some zero columns. Since (\mathbf{K}_{pp})_{h} is, in the same basis, diagonal with the bulk modulus -\kappa^{-1} as diagonal elements and the corresponding right-hand-side is a zero vector, the solution for the spurious pressure mode values is zero (see also Example 4.39).
A different observation is that the coefficient matrix in (4.201) contains a large bulk modulus which, when \kappa^{-1} is close to zero, results in ill-conditioning—but this ill-conditioning is observed whether or not spurious pressure modes are present.
The spurious pressure modes can, however, have a drastic effect when nonzero displacements are prescribed. In this case, we recognize that the right-hand side corresponding to the pressure degrees of freedom may not be zero (see Section 4.2.2 on how nonzero displacements are imposed), and a large spurious pressure may be generated.
Clearly, a reliable element should not lock and ideally should not lead to any spurious pressure mode in any chosen mesh.
The elements listed in Tables 4.6 and 4.7 are of such a nature—except for the 4/1 two-dimensional u/p element (and the analogous 8/1 three-dimensional element). Using the 4/1 element, specific meshes with certain boundary conditions exhibit a spurious pressure mode, and the 4/1 element does not satisfy the inf-sup condition (4.183) unless used in special geometric arrangements of macroelements (see P. Le Tallec and V. Ruas [A] for an example). However, because of its simplicity, the 4/1 element is quite widely used in practice. We examine this element in more detail in the following example.
EXAMPLE 4.38: Consider the finite element discretization of 4/1 elements shown in Fig. E4.38 and show that the spurious checkerboard mode of pressure indicated in the figure exists.
We note that for this model all tangential displacements on the boundary are set to zero. In order to show that the pressure distribution indicated in Fig. E4.38 corresponds to a spurious pressure mode, we need to prove that (4.196) holds. Consider a single element as shown in Fig. E4.38(a). We have
\int_ {\mathrm{Vol}} p ^ {e _ {i}} \operatorname{div} \mathbf {v} _ {h} ^ {e} d \mathrm{Vol} = p ^ {e _ {i}} \left[ \begin{array}{l l l l l l l l l l l} 1 & - 1 & - 1 & 1 & \vdots & 1 & 1 & - 1 & - 1 \end{array} \right] \hat {\mathbf {u}}
where p^{e_{i}} is the constant pressure in the element e_{i} .
If a patch of four adjacent elements is then considered, we note that for the displacement u_{i} shown in Fig. E4.38(b) we have
\int_ {\mathrm{Vol}} p \operatorname{div} \mathbf {v} _ {h} d \mathrm{Vol} = \left[ p ^ {e _ {1}} (1) + p ^ {e _ {2}} (1) + p ^ {e _ {3}} (- 1) + p ^ {e _ {4}} (- 1) \right] u _ {i} = 0 \tag {a}
provided the pressure distribution corresponds to p^{e_1} = -p^{e_2} = p^{e_3} = -p^{e_4} . Similarly, for any displacement v_i we have
\int_ {\mathrm{Vol}} p \operatorname{div} \mathbf {v} _ {h} d \mathrm{Vol} = \left[ p ^ {e _ {1}} (- 1) + p ^ {e _ {2}} (1) + p ^ {e _ {3}} (1) + p ^ {e _ {4}} (- 1) \right] v _ {i} = 0 \tag {b}
text_image
2 2 y v₁ 1 u₁ 2 pσᵢ x 3 4
(a) Single element
text_image
2 2
(c) 4 \times 4 mesh of equal square elements
text_image
2 v_i u_j j p^e1 p^e4 v_i i u_i p^e2 p^e3 2
(b) Patch of four equal elements
| + | - | + | - |
| - | + | - | + |
| + | - | + | - |
| - | + | - | + |
(d) Checkerboard pressure distribution. + and - mean +Δp and -Δp, where Δp is an arbitrary value.
Figure E4.38 4/1 elements
For the normal displacement v_{j} on an edge of the patch, we similarly obtain
\int_ {\mathrm{Vol}} p \operatorname{div} \mathbf {v} _ {h} d \mathrm{Vol} = [ p ^ {e _ {1}} (1) + p ^ {e _ {2}} (1) ] v _ {j} = 0 \tag {c}
On the other hand, for a tangential displacement u_{j} , the integral
\int_ {\mathrm{Vol}} p \operatorname{div} \mathbf {v} _ {h} d \mathrm{Vol} \neq 0
However, in the model in Fig. E4.38(c) all tangential displacements are constrained to zero. Hence, by superposition, using expressions (a) to (c), the relation (4.196) is satisfied for any nodal point displacements when the pressure distribution is the indicated checkerboard pressure.
Note that the same checkerboard pressure distribution is also a spurious pressure mode when more nodal point displacements than those given in Fig. E4.38(c) are constrained to zero. Also note that the (assumed) pressure distribution in Fig. E4.38(d) cannot be obtained by any nodal point displacements, hence this pressure distribution does not correspond to an element in P_{h}(D_{h}) .
In the above example, we showed that a spurious pressure mode is present when the 4/1 element is used in discretizations of equal-size square elements with certain boundary
conditions. The spurious pressure mode no longer exists when nonhomogeneous meshes are employed or at least one tangential displacement on the surface is released to be free.
Consider now that a force is applied to any one of the free degrees of freedom in Fig. E4.38(c). The solution is then obtained by solving (4.201) and, as pointed out before, the spurious pressure mode will not enter the solution (it will not be observed).
The spurious pressure mode, however, has a very significant effect on the calculated stresses when, for example, one tangential boundary displacement is prescribed to be nonzero while all other tangential boundary displacements are kept at zero. ^{18} In this case the prescribed nodal point displacement results in a nonzero forcing vector for the pressure degrees of freedom, and the spurious pressure mode is excited. Hence, in practice, it is expedient to not constrain all tangential nodal point displacements on the body considered.
Let us conclude this section by considering the following example because it illustrates, in a simple manner, some of the important general observations we have made.
EXAMPLE 4.39: ^{19} Assume that the governing equations (4.187) are
\left[ \begin{array}{c c c c c} \alpha_ {1} & 0 & 0 & \beta_ {1} & 0 \\ 0 & \alpha_ {2} & 0 & 0 & \beta_ {2} \\ 0 & 0 & \alpha_ {3} & 0 & 0 \\ \hline \beta_ {1} & 0 & 0 & 0 & 0 \\ 0 & \beta_ {2} & 0 & 0 & 0 \end{array} \right] \left[ \begin{array}{l} u _ {1} \\ u _ {2} \\ u _ {3} \\ \vdots \\ p _ {1} \\ p _ {2} \end{array} \right] = \left[ \begin{array}{l} r _ {1} \\ r _ {2} \\ r _ {3} \\ \vdots \\ g _ {1} \\ g _ {2} \end{array} \right] \tag {a}
Of course, such simple equations are not obtained in practical finite element analysis, but the essential ingredients are those of the general equations (4.187). We note that the coefficient matrix corresponds to a fully incompressible material condition and that the entries g_{1} and g_{2} correspond to prescribed boundary displacements.
These equations can also be written as
\alpha_ {i} u _ {i} + \beta_ {i} p _ {i} = r _ {i}; \quad \beta_ {i} u _ {i} = g _ {i}; \quad i = 1, 2; \quad \alpha_ {3} u _ {3} = r _ {3}
Assume that \alpha_{i} > 0 for all i (as we would have in practice). Then, u_{3} = r_{3} / \alpha_{3} , and we need only consider the typical equations
\alpha u + \beta p = r; \quad \beta u = g \tag {b}
(where we have dropped the subscript i).
When the material is considered almost incompressible, u_{3} is unchanged but (b) becomes
\alpha u _ {\epsilon} + \beta p _ {\epsilon} = r; \quad \beta u _ {\epsilon} - \epsilon p _ {\epsilon} = g \tag {c}
where \epsilon = 1 / \kappa ( \epsilon is very small when the bulk modulus \kappa is very large) and u_{\epsilon}, p_{\epsilon} is the solution sought. Equations (c) give
u _ {\epsilon} = \frac {\epsilon r + \beta g}{\epsilon \alpha + \beta^ {2}}; \quad p _ {\epsilon} = \frac {\beta r - \alpha g}{\epsilon \alpha + \beta^ {2}} \tag {d}
We can now make the following observations.
First, we consider the case of a spurious pressure mode, i.e., \beta = 0 .
Case i: \beta = g = 0
This case corresponds to a spurious pressure mode and zero prescribed displacements.
The solution of (b) gives u = r / \alpha , with p undetermined.
The solution of (c) gives u_{\epsilon} = r/\alpha , p_{\epsilon} = 0 .
Hence, we notice that the use of a finite bulk modulus allows us to solve the equations and suppresses the spurious pressure.
Case ii: \beta = 0, g \neq 0
This case corresponds to a spurious pressure mode and nonzero prescribed displacements (corresponding to this mode).
Now (b) has no solution for u and p .
The solution of (c) is u_{\epsilon} = r / \alpha, p_{\epsilon} = -g / \epsilon .
Hence, the spurious pressure becomes large as \kappa increases.
Next we consider the case of \beta very small.
Hence, we have no spurious pressure mode. Of course, the inf-sup condition is not passed if \beta \to 0 .
Case iii: \beta is small
Let us also assume that g = 0.
Now (b) gives the solution u = 0, p = r / \beta .
The solution of (c) is u_{\epsilon} \rightarrow 0 and p_{\epsilon} \rightarrow r/\beta for \epsilon \rightarrow 0 ( \beta fixed, and hence we have \beta^{2} \gg \epsilon\alpha ), which is consistent with the solution of (b). Hence, the displacement approaches zero and the pressure becomes large when \beta is small and the bulk modulus increases. Of course, we test for this behavior with the inf-sup condition. For an actual finite element solution, this observation may be interpreted as taking a fixed mesh ( \beta is fixed) and increasing \kappa . The result is that the pressure in the mode for which \beta is small increases while the displacements in this mode decrease.
However, (c) also gives u_{\epsilon} \rightarrow r/\alpha and p_{\epsilon} \rightarrow 0 for \beta \rightarrow 0 ( \epsilon fixed, and hence we have \beta^{2} \ll \epsilon\alpha ), which is the behavior noted earlier in Case i. For an actual finite element solution this observation may be interpreted as taking a fixed \kappa and increasing the fineness of the mesh. As \beta is decreased as a result of mesh refinement, the pressure corresponding to this mode becomes small. Hence, the behavior of this pressure mode is when \beta is sufficiently small (which may mean a very fine mesh when \kappa is large) like the behavior of a spurious mode.
4.5.6 The Inf-Sup Test
The results of analytical studies of the inf-sup characteristics of various displacement/pressure elements are summarized in Tables 4.6 and 4.7 (see also F. Brezzi and M. Fortin [A]). However, an analytical proof of whether the inf-sup condition is satisfied by a specific element can be difficult, and for this reason a numerical test is valuable. Such a test can be applied to newly proposed elements and also to discretizations with elements of distorted geometries (recall that analytical studies assume homogeneous meshes of square elements). Of course, a numerical test cannot be completely affirmative (as an analytical proof is), but if a properly designed numerical test is passed, the formulation is very likely to be effective. The same idea is used when performing the patch test only in numerical form (to study incompatible displacement formulations and the effect of element geometric distortions) because an analytical evaluation is not achieved (see Section 4.4.1).
In the following discussion we present the numerical inf-sup test proposed by D. Chapelle and K. J. Bathe [A].
First consider the u / p formulation. In this case the inf-sup condition (4.183) can be written in the form
\inf _ {\mathbf {w} _ {h} \in \mathcal {V} _ {h}} \sup _ {\mathbf {v} _ {h} \in \mathcal {V} _ {h}} \frac {\int_ {\mathrm{Vol}} P _ {h} (\operatorname{div} \mathbf {w} _ {h}) \operatorname{div} \mathbf {v} _ {h} d \operatorname{Vol}}{\| P _ {h} (\operatorname{div} \mathbf {w} _ {h}) \| \| \mathbf {v} _ {h} \|} \geq \beta > 0 \tag {4.203}
\inf _ {\mathbf {w} _ {h} \in V _ {h}} \sup _ {\mathbf {v} _ {h} \in V _ {h}} \frac {b ^ {\prime} (\mathbf {w} _ {h} , \mathbf {v} _ {h})}{[ b ^ {\prime} (\mathbf {w} _ {h} , \mathbf {w} _ {h}) ] ^ {1 / 2} \| \mathbf {v} _ {h} \|} \geq \beta > 0 \tag {4.204}
\text { where } \quad b ^ {\prime} (\mathbf {w} _ {h}, \mathbf {v} _ {h}) = \int_ {\mathrm{Vol}} P _ {h} (\mathrm{div} \mathbf {w} _ {h}) P _ {h} (\mathrm{div} \mathbf {v} _ {h}) d \mathrm{Vol} = \int_ {\mathrm{Vol}} P _ {h} (\mathrm{div} \mathbf {w} _ {h}) \mathrm{div} \mathbf {v} _ {h} d \mathrm{Vol} \tag {4.205}
The relation (4.204) is in matrix form
\inf _ {\mathbf {w} _ {h}} \sup _ {\mathbf {v} _ {h}} \frac {\mathbf {W} _ {h} ^ {T} \mathbf {G} _ {h} \mathbf {V} _ {h}}{\left[ \mathbf {W} _ {h} ^ {T} \mathbf {G} _ {h} \mathbf {W} _ {h} \right] ^ {1 / 2} \left[ \mathbf {V} _ {h} ^ {T} \mathbf {S} _ {h} \mathbf {V} _ {h} \right] ^ {1 / 2}} \geq \beta > 0 \tag {4.206}
where W_{h} and V_{h} are vectors of the nodal displacement values corresponding to w_{h} and v_{h} , and G_{h} , S_{h} are matrices corresponding to the operator b' and the norm \|\cdot\|_{V} , respectively. The matrices G_{h} and S_{h} are, respectively, positive semidefinite and positive definite (for the problem we consider, see Section 4.5.1).
EXAMPLE 4.40: In Example 4.34 we calculated the matrix G_{h} of a 4/1 element. Now also establish the matrix S_{h} of this element.
To evaluate S_{h} we recall that the norm of w is given by [see (4.153)]
\| \mathbf {w} \| _ {V} ^ {2} = \sum_ {i, j} \left\| \frac {\partial w _ {i}}{\partial x _ {j}} \right\| _ {L ^ {2} (\mathrm{Vol})} ^ {2}
Hence, for our case
\| \mathbf {w} _ {h} \| _ {V} ^ {2} = \int_ {- 1} ^ {+ 1} \int_ {- 1} ^ {+ 1} \left[ \left(\frac {\partial u}{\partial x}\right) ^ {2} + \left(\frac {\partial u}{\partial y}\right) ^ {2} + \left(\frac {\partial v}{\partial x}\right) ^ {2} + \left(\frac {\partial v}{\partial y}\right) ^ {2} \right] d x d y \tag {a}
where u, v are the components w_{i}, i = 1, 2 .
Let us order the nodal point displacements in \hat{u} as in Example 4.34,
\hat {\mathbf {u}} ^ {T} = \left[ \begin{array}{l l l l l l l l l} u _ {1} & u _ {2} & u _ {3} & u _ {4} & \vdots & v _ {1} & v _ {2} & v _ {3} & v _ {4} \end{array} \right]
By definition, \| \mathbf{w}_h\| _V^2 = \hat{\mathbf{u}}^T\mathbf{S}_h\hat{\mathbf{u}} . Also, we have
\frac {\partial u}{\partial x} = \sum_ {i = 1} ^ {4} h _ {i, x} u _ {i}; \quad \frac {\partial u}{\partial y} = \sum_ {i = 1} ^ {4} h _ {i, y} u _ {i} \tag {b}
and we write in (a)
\left(\frac {\partial u}{\partial x}\right) ^ {2} = \left(\frac {\partial u}{\partial x}\right) ^ {T} \left(\frac {\partial u}{\partial x}\right) \tag {c}
\left(\frac {\partial u}{\partial y}\right) ^ {2} = \left(\frac {\partial u}{\partial y}\right) ^ {T} \left(\frac {\partial u}{\partial y}\right)
Substituting from (c) and (b) into (a) we obtain
\mathrm{S} _ {h} (1, 1) = \int_ {- 1} ^ {+ 1} \int_ {- 1} ^ {+ 1} \left[ \left(h _ {1, x}\right) ^ {2} + \left(h _ {1, y}\right) ^ {2} \right] d x d y = \frac {2}{3}
\mathrm{S} _ {h} (1, 2) = \int_ {- 1} ^ {+ 1} \int_ {- 1} ^ {+ 1} \left[ h _ {1, x} h _ {2, x} + h _ {1, y} h _ {2, y} \right] d x d y = - \frac {1}{6}
and so on.