김경종
411 lines
20 KiB
Markdown
411 lines
20 KiB
Markdown
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TABLE 4.7 Various effective u/p-c elements (displacements and pressure between elements are continuous and all elements satisfy the inf-sup condition) $^{\dagger}$
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<table><tr><td rowspan="2">Element</td><td colspan="3">Nodal points</td><td rowspan="2">Remarks</td></tr><tr><td>2-D element</td><td colspan="2">3-D element</td></tr><tr><td> $Q_{2} - Q_{1}$ in 2-D: 9/4-cin 3-D: 27/8-c</td><td></td><td>Only the visible 19 of the total 27 nodes are shown</td><td>Only the 8 non-visible nodes are shown</td><td>See P. Hood and C. Taylor [A]. The first member of the $Q_{n} - Q_{n-1}$ family of quadrilateral elements, $n \geq 2$ .</td></tr><tr><td></td><td></td><td></td><td>Node in center of element</td><td></td></tr></table>
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See P. Hood and C. Taylor [A]. The first member of the $P_{n} - P_{n - 1}$ family of triangular elements, $n\geq 2$
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See D. N. Arnold, F. Brezzi, and M. Fortin [A]. $P_{1}^{+}$ denotes the space of polynomials $A$ enriched by the cubic bubble. Also referred to as MINI elements.
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<details>
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<summary>natural_image</summary>
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Geometric diagram of a polyhedron with solid and dashed edges, no text or symbols present
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</details>
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<details>
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<summary>natural_image</summary>
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Geometric diagram of a quadrilateral with internal diagonals and a central dot (no text or labels)
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</details>
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<details>
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<summary>natural_image</summary>
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Simple geometric triangle diagram with five vertices and connecting lines (no text or symbols)
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</details>
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<details>
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<summary>natural_image</summary>
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Simple geometric diagram of a triangle with three vertices and a central dot (no text or labels)
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</details>
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$P_{2} - P_{1}$ in 2-D: 6/3-c in 3-D: 10/4-c
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$P_{1}^{+} - P_{1}$ in 2-D: 4/3-c in 3-D: 5/4-c
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- Node with displacement variables
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© Node with displacement and pressure variables
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$^{\dagger}$ For the interpolation functions, see Figs. 4.13, 5.4, 5.5, 5.11, and 5.13.
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We may ask whether in practice it is really important to satisfy the inf-sup condition, that is, whether perhaps this condition is too strong and elements that do not satisfy it can still be used reliably. Our experience is that if the inf-sup condition is satisfied, the element will be, for the interpolations used, as effective as we can reasonably expect and in that sense optimal. For example, the 9/3 element for plane strain analysis in Table 4.6 is based on a parabolic interpolation of displacements and a linear interpolation of pressure. The element does not lock, and the order of convergence of displacements is always $o(h^{3})$ , and of stresses, $o(h^{2})$ , which is surely the best behavior we can obtain with the interpolations used.
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On the other hand, if the inf-sup condition is not satisfied, the element will not always display for all analysis problems (pertaining to the mathematical model considered) the convergence characteristics that we would expect and indeed require in practice. The element is therefore not robust and reliable.
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Since the inf-sup condition is of great fundamental importance, we present in the following section a derivation that although not mathematically complete does yield valuable insight. In this discussion we will also encounter and briefly exemplify the ellipticity condition. For a mathematically complete derivation of the ellipticity and inf-sup conditions and many more details, we refer the reader to the book by F. Brezzi and M. Fortin [A].
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In the derivation in the next section we examine the problem of incompressible elasticity, but our considerations are also directly applicable to the problem of incompressible fluid flow, and as shown in Section 4.5.7, to the formulations of structural elements.
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# 4.4.4 Exercises
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4.33. Use the four-node and eight-node shell elements available in a finite element program and perform the patch tests in Fig. 4.17.
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4.34. Consider the three-dimensional eight-node element shown. Design the patch test and identify analytically whether it is passed for the element.
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<details>
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<summary>text_image</summary>
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y
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8
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2
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</details>
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$$
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u = \sum_ {i = 1} ^ {8} h _ {i} u _ {i} + \alpha_ {1} \phi_ {1} + \alpha_ {2} \phi_ {2} + \alpha_ {3} \phi_ {3}
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$$
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$$
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v = \sum_ {i = 1} ^ {8} h _ {i} v _ {i} + \alpha_ {4} \phi_ {1} + \alpha_ {5} \phi_ {2} + \alpha_ {6} \phi_ {3}
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$$
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$$
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w = \sum_ {i = 1} ^ {8} h _ {i} w _ {i} + \alpha_ {7} \phi_ {1} + \alpha_ {8} \phi_ {2} + \alpha_ {9} \phi_ {3}
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$$
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$$
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h _ {i} = \frac {1}{8} (1 + x _ {i} x) (1 + y _ {i} y) (1 + z _ {i} z)
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$$
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$$
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\phi_ {1} = 1 - x ^ {2}; \phi_ {2} = 1 - y ^ {2}; \phi_ {3} = 1 - z ^ {2}
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$$
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4.35. Consider the Hu-Washizu functional $\Pi_{\mathbf{HW}}$ in (4.114) and derive in detail the equations (4.116) to (4.121).
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4.36. The following functional is referred to as the Hellinger-Reissner functional $^{17}$
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$$
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\begin{array}{l} \Pi_ {\mathrm{HR}} (\mathbf {u}, \tau) = \int_ {V} - \frac {1}{2} \tau^ {T} \mathbf {C} ^ {- 1} \tau d V + \int_ {V} \tau^ {T} \partial_ {\epsilon} \mathbf {u} d V \\ - \int_ {V} \mathbf {u} ^ {T} \mathbf {f} ^ {B} d V - \int_ {S _ {f}} \mathbf {u} ^ {S _ {f} T} \mathbf {f} ^ {S _ {f}} d S - \int_ {S _ {u}} \mathbf {f} ^ {S _ {u} T} \left(\mathbf {u} ^ {S _ {u}} - \mathbf {u} _ {p}\right) d S \\ \end{array}
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$$
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where the prescribed (not to be varied) quantities are $\mathbf{f}^B$ in $V$ , $\mathbf{u}_p$ on $S_u$ , and $\mathbf{f}^{S_f}$ on $S_f$ .
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Derive this functional from the Hu-Washizu functional by imposing $\epsilon = \mathbf{C}^{-1}\tau$ . Then invoke the stationarity of $\Pi_{\mathrm{HR}}$ and establish all remaining differential conditions for the volume and surface of the body.
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4.37. Consider the functional
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$$
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\Pi_ {1} = \Pi - \int_ {S _ {u}} \mathbf {f} ^ {S _ {u} T} (\mathbf {u} ^ {S _ {u}} - \mathbf {u} _ {p}) d S
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$$
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where $\Pi$ is given in (4.109) and $\mathbf{u}_{\rho}$ are the displacements to be prescribed on the surface $S_{u}$ . Hence, the vector $\mathbf{f}^{S_{u}}$ represents the Lagrange multipliers (surface tractions) used to enforce the surface displacement conditions. Invoke the stationarity of $\Pi_{1}$ and show that the Lagrange multiplier term will enforce the displacement boundary conditions on $S_{u}$ .
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4.38. Consider the three-node truss element in Fig. E4.29. Use the Hu-Washizu variational principle and establish the stiffness matrices for the following assumptions:
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(a) Parabolic displacement, linear strain, and constant stress
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(b) Parabolic displacement, constant strain, and constant stress
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Discuss your results in terms of whether the choices of interpolations are sensible (see Example 4.29).
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4.39. Show that the following stress-strain expressions of an isotropic material are equivalent.
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$$
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\tau_ {i j} = \kappa \epsilon_ {V} \delta_ {i j} + 2 G \epsilon_ {i j} ^ {\prime} \tag {a}
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$$
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$$
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\tau_ {i j} = C _ {i j r s} \epsilon_ {r s} \tag {b}
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$$
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$$
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\boldsymbol {\tau} = \mathbf {C} \boldsymbol {\epsilon} \tag {c}
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$$
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where $\kappa$ is the bulk modulus, $G$ is the shear modulus,
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$$
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\kappa = \frac {E}{3 (1 - 2 \nu)}; \quad G = \frac {E}{2 (1 + \nu)}
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$$
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E is Young's modulus, $\nu$ is Poisson's ratio, $\epsilon_{\nu}$ is the volumetric strain, and $\epsilon_{ij}^{\prime}$ are the deviatoric strain components,
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$$
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\epsilon_ {V} = \epsilon_ {k k}; \quad \epsilon_ {i j} ^ {\prime} = \epsilon_ {i j} - \frac {\epsilon_ {V}}{3} \delta_ {i j}
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$$
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Also, $C_{ijrs} = \lambda \delta_{ij}\delta_{rs} + \mu (\delta_{ir}\delta_{js} + \delta_{is}\delta_{jr})$
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$^{17}$ This functional is sometimes given in a different form by applying the divergence theorem to the second term.
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<!-- source-page: 315 -->
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where $\lambda$ and $\mu$ are the Lamé constants,
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$$
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\lambda = \frac {E \nu}{(1 + \nu) (1 - 2 \nu)}; \quad \mu = \frac {E}{2 (1 + \nu)}
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$$
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In (a) and (b) tensorial quantities are used, whereas in (c) the vector of strains contains the engineering shear strains (which are equal to twice the tensor components; e.g., $\gamma_{xy} = \epsilon_{12} + \epsilon_{21}$ ). Also, the stress-strain matrix C in (c) is given in Table 4.3.
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4.40. Identify the order of pressure interpolation that should be used in the u/p formulation in order to obtain the same stiffness matrix as in the pure displacement formulation. Consider the following elements of $2 \times 2$ geometry.
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(a) Four-node element in plane strain
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(b) Four-node element in axisymmetric conditions
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(c) Nine-node element in plane strain.
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4.41. Consider the 4/1 element in Example 4.32 and assume that the displacement boundary condition to be imposed is $u_{1} = \overline{u}$ . Show formally that imposing this boundary condition prior to or after the static condensation of the pressure degree of freedom, yields the same element contribution to the stiffness matrix of the assemblage.
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4.42. Consider the axisymmetric $4/1 u/p$ element shown. Construct the matrices $\mathbf{B}_D, \mathbf{B}_V, \mathbf{C}'$ , and $\mathbf{H}_p$ for this element.
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<details>
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<summary>text_image</summary>
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€
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1
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2
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</details>
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4.43. Consider the 4/3-c element in plane strain conditions shown. Formulate all displacement and strain interpolation matrices for this element (see Table 4.7).
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<details>
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<summary>text_image</summary>
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y, v
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x, u
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</details>
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4.44. Consider the 9/3 plane strain u/p element shown. Calculate the matrix $K_{pp}$ .
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<details>
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<summary>text_image</summary>
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y
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</details>
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Young's modulus E
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Poisson's ratio v = 0.49
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4.45. Consider the plate with the circular hole shown. Use a finite element program to solve for the stress distribution along section AA for the two cases of Poisson's ratios $\nu = 0.3$ and $\nu = 0.499$ . Assess the accuracy of your results by means of an error measure. (Hint: For the analysis with $\nu = 0.499$ , the 9/3 element is effective.)
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<details>
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<summary>text_image</summary>
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A
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100 mm
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100 mm
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r = 40 mm
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p
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100 mm
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100 mm
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A
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</details>
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Plane strain condition
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Young's modulus E = 200,000 MPa
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4.46. The static response of the thick cylinder shown is to be calculated with a finite element program.
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<details>
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<summary>text_image</summary>
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f
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f = force per unit length
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20 mm
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10 mm
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30 mm
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E = 200,000 MPa
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v = 0.499
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</details>
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Use idealizations based on the following elements to analyze the cylinder.
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(a) Four-node displacement-based element
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(b) Nine-node displacement-based element
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(c) $4 / 1u / p$ element.
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(d) 9/3 u/p element.
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In each case use a sequence of meshes and identify the convergence rate of the strain energy.
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# 4.5 THE INF-SUP CONDITION FOR ANALYSIS OF INCOMPRESSIBLE MEDIA AND STRUCTURAL PROBLEMS
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As we pointed out in the previous section, it is important that the finite element discretization for the analysis of almost, and of course totally, incompressible media satisfy the inf-sup condition. The objective in this section is to present this condition. We first consider the pure displacement formulation for the analysis of solids and then the displacement/pres-
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sure formulations. Finally, we also briefly discuss the inf-sup condition as applicable to structural elements.
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In our discussion we apply the displacement and displacement/pressure formulations to a solid medium. However, the basic observations and conclusions are also directly applicable to the solution of incompressible fluid flows if velocities are used instead of displacements (see Section 7.4).
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# 4.5.1 The Inf-Sup Condition Derived from Convergence Considerations
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We want to solve a general linear elasticity problem (see Section 4.2.1) in which a body is subjected to body forces $f^{B}$ , surface tractions $f^{S_{f}}$ on the surface $S_{f}$ , and displacement boundary conditions $u^{S_{u}}$ on the surface $S_{u}$ . Without loss of generality of the conclusions that we want to reach in this section, we can assume that the prescribed displacements $u^{S_{u}}$ and prescribed tractions $f^{S_{f}}$ are zero. Of course, we assume that the body is properly supported, so that no rigid body motions are possible. We can then write our analysis problem as a problem of minimization,
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$$
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\min _ {\mathbf {v} \in V} \left\{\frac {1}{2} a (\mathbf {v}, \mathbf {v}) + \frac {\kappa}{2} \int_ {\mathrm{Vol}} (\operatorname{div} \mathbf {v}) ^ {2} d \mathrm{Vol} - \int_ {\mathrm{Vol}} \mathbf {f} ^ {B} \cdot \mathbf {v} d \mathrm{Vol} \right\} \tag {4.151}
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$$
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where using indicial notation and tensor quantities (see Sections 4.3.4 and 4.4.3),
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$$
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a (\mathbf {u}, \mathbf {v}) = 2 G \int_ {\mathrm{Vol}} \sum_ {i, j} ^ {3} \epsilon_ {i j} ^ {\prime} (\mathbf {u}) \epsilon_ {i j} ^ {\prime} (\mathbf {v}) d \mathrm{Vol}
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$$
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$$
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\epsilon_ {i j} ^ {\prime} (\mathbf {u}) = \epsilon_ {i j} (\mathbf {u}) - \frac {1}{3} \operatorname{div} \mathbf {u} \delta_ {i j} \tag {4.152}
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$$
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$$
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\epsilon_ {i j} (\mathbf {u}) = \frac {1}{2} \left(\frac {\partial u _ {i}}{\partial x _ {j}} + \frac {\partial u _ {j}}{\partial x _ {i}}\right); \quad \operatorname{div} \mathbf {v} = v _ {i, i}
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$$
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where $\kappa = E / [3(1 - 2\nu)]$ (bulk modulus), $G = E / [2(1 + \nu)]$ (shear modulus), $E =$ Young's modulus, $\nu =$ Poisson's ratio.
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$$
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V = \left\{\mathbf {v} \mid \frac {\partial v _ {i}}{\partial x _ {j}} \in L ^ {2} (\mathrm{Vol}), i, j = 1, 2, 3; v _ {i} | _ {s _ {u}} = 0, i = 1, 2, 3 \right\}
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$$
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In these expressions we use the notation defined earlier (see Section 4.3) and we denote by "Vol" the domain over which we integrate so as to avoid any confusion with the vector space V. Also, we use for the vector v and scalar q the norms
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$$
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\| \mathbf {v} \| _ {V} ^ {2} = \sum_ {i, j} \left\| \frac {\partial v _ {i}}{\partial x _ {j}} \right\| _ {L ^ {2} (\mathrm{Vol})} ^ {2}; \quad \| q \| _ {0} ^ {2} = \| q \| _ {L ^ {2} (\mathrm{Vol})} ^ {2} \tag {4.153}
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$$
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where the vector norm $\| \cdot \|_V$ is somewhat easier to work with but is equivalent to the Sobolev norm $\| \cdot \|_1$ defined in (4.76) (by the Poincaré-Friedrichs inequality).
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In the following discussion we will not explicitly give the subscripts on the norms but always imply that a vector $\mathbf{w}$ has norm $\| \mathbf{w}\|_{\nu}$ and a scalar $\gamma$ has norm $\| \gamma \|_0$ .
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Let u be the minimizer of (4.151) (i.e., the exact solution to the problem) and let $V_{h}$ be a space of a sequence of finite element spaces that we choose to solve the problem. These spaces are defined in (4.84). Of course, each discrete problem,
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$$
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\min _ {\mathbf {v} _ {h} \in V _ {h}} \left\{\frac {1}{2} a \left(\mathbf {v} _ {h}, \mathbf {v} _ {h}\right) + \frac {\kappa}{2} \int_ {\text { Vol }} \left(\operatorname{div} \mathbf {v} _ {h}\right) ^ {2} d \text { Vol } - \int_ {\text { Vol }} \mathbf {f} ^ {B} \cdot \mathbf {v} _ {h} d \text { Vol } \right\} \tag {4.154}
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$$
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has a unique finite element solution $u_{h}$ . We considered the properties of this solution in Section 4.3.4, and in particular we presented the properties (4.95) and (4.101). However, we also stated that the constants c in these relations are dependent on the material properties. The important point now is that when the bulk modulus $\kappa$ is very large, the relations (4.95) and (4.101) are no longer useful because the constants are too large. Therefore, we want our finite element space $V_{h}$ to satisfy another property, still of the form (4.95) but in which the constant c, in addition to being independent of h, is also independent of $\kappa$ .
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To state this new desired property, let us first define the “distance” between the exact solution u and the finite element space $V_{h}$ (see Fig. 4.22),
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$$
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d (\mathbf {u}, V _ {h}) = \inf _ {\mathbf {v} _ {h} \in V _ {h}} \| \mathbf {u} - \mathbf {v} _ {h} \| = \| \mathbf {u} - \tilde {\mathbf {u}} _ {h} \| \tag {4.155}
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$$
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where $\tilde{u}_{h}$ is an element in $V_{h}$ but is in general not the finite element solution.
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# The Basic Requirements
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In engineering practice, the bulk modulus $\kappa$ may vary from values of the order of G to very large values, and indeed to infinity when complete incompressibility is considered. Our objective is to use finite elements that are uniformly effective irrespective of what value $\kappa$
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<details>
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<summary>text_image</summary>
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||u - u_h||
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d(u, V_h) = ||u - \tilde{u}_h||
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u
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u_h
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\tilde{u}_h
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V_h
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</details>
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Figure 4.22 Schematic representation of solutions and distances; for optimal convergence $\|u - u_{h}\| \leq c d(u, V_{h})$ with c independent of h and $\kappa$ .
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<!-- source-page: 320 -->
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takes. Mathematically, therefore, our purpose is to find conditions on $V_h$ such that
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$$
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\| \mathbf {u} - \mathbf {u} _ {h} \| \leq c d (\mathbf {u}, V _ {h})
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$$
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with a constant $c$ independent of $h$ and $\kappa$ .
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(4.156)
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These conditions shall guide us in our choice of effective finite elements and discretizations.
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The inequality (4.156) means that the distance between the continuous solution u and the finite element solution $u_{h}$ will be smaller than a (reasonably sized) constant c times $d(\mathbf{u}, V_{h})$ and that this relationship will be satisfied with the same constant c irrespective of the bulk modulus used. Note that this independence of c from the bulk modulus is the key property we did not have in Section 4.3.4 when we derived a relation such as (4.156) [see (4.95)].
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Assume that the condition (4.156) holds (with a reasonably sized constant c). Then if $d(\mathbf{u}, V_{h})$ is $o(h^{k})$ , we know that $\|u - u_{h}\|$ is also $o(h^{k})$ , and since c is reasonably sized and independent of $\kappa$ , we will in fact observe the same solution accuracy and improvement in accuracy as h is decreased irrespective of the bulk modulus in the problem. In this case the finite element spaces have good approximation properties for any value of $\kappa$ , and the finite element discretization is reliable (see Section 1.3).
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The relationship in (4.156) expresses our fundamental requirement for the finite element discretization, and finite element formulations that satisfy (4.156) do not lock (see Section 4.4.3). In the following discussion, we write (4.156) only in forms with which we can work more easily in choosing effective finite elements. One of these forms uses an inf-sup value and is the celebrated inf-sup condition.
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To proceed further, we define the spaces $K$ and $D$ ,
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$$
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K (q) = \{\mathbf {v} \mid \mathbf {v} \in V, \operatorname{div} \mathbf {v} = q \} \tag {4.157}
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$$
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$$
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D = \{q \mid q = \operatorname{div} \mathbf {v} \text { for some } \mathbf {v} \in V \} \tag {4.158}
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$$
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and the corresponding spaces for our discretizations,
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$$
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K _ {h} (q _ {h}) = \{\mathbf {v} _ {h} \mid \mathbf {v} _ {h} \in V _ {h}, \operatorname{div} \mathbf {v} _ {h} = q _ {h} \} \tag {4.159}
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$$
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$$
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D _ {h} = \{q _ {h} \mid q _ {h} = \operatorname{div} \mathbf {v} _ {h} \text { for some } \mathbf {v} _ {h} \in V _ {h} \} \tag {4.160}
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$$
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Hence the space $K_{h}(q_{h})$ , for a given $q_{h}$ , corresponds to all the elements $v_{h}$ in $V_{h}$ that satisfy $\operatorname{div} v_{h} = q_{h}$ . Also, the space $D_{h}$ corresponds to all the elements $q_{h}$ with $q_{h} = \operatorname{div} v_{h}$ that are reached by the elements $v_{h}$ in $V_{h}$ ; that is, for any $q_{h}$ an element of $D_{h}$ there is at least one element $v_{h}$ in $V_{h}$ such that $q_{h} = \operatorname{div} v_{h}$ . Similar thoughts are applicable to the spaces K and D.
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We recall that when $\kappa$ is large, the quantity $\| \operatorname{div} \mathbf{u}_h\|$ will be small; the larger $\kappa$ , the smaller $\| \operatorname{div} \mathbf{u}_h\|$ , and it is difficult to obtain an accurate pressure prediction $p_h = -\kappa \operatorname{div} \mathbf{u}_h$ . In the limit $\kappa \to \infty$ we have $\operatorname{div} \mathbf{u}_h = 0$ , but the pressure $p_h$ is still finite (and of course of order of the applied tractions) and therefore $\kappa(\operatorname{div} \mathbf{u}_h)^2 = 0$ .
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