MultiPhysicsVault/.raw/ConceptsApplicationsFiniteElementAnalysis/ConceptsApplicationsFiniteElementAnalysis_021.md

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3 η 7 4 8 2 1 ξ 6 5

Figure 6.7-1. Linear solid (eight-node brick) element. Also called a “trilinear” element.

In structural mechanics we deal with displacements u , v , and w , which are parallel to x , y , and z directions, respectively. Matrix [B] is evaluated by expanded forms of Eqs. 6.3-17 through 6.3-19. As expanded, Eq. 6.3-19 has nine rows, Eq. 6.3-18 contains a 9 by 9 square matrix, and Eq. 6.3-17 states the strain-displacement relations (Eqs. 1.5-6) as

\left\{ \begin{array}{l} \epsilon_ {x} \\ \epsilon_ {y} \\ \epsilon_ {z} \\ \gamma_ {x y} \\ \gamma_ {y z} \\ \gamma_ {z x} \end{array} \right\} = [ \mathbf {H} ] \left\{ \begin{array}{l} u _ {, x} \\ u _ {, y} \\ u _ {, z} \\ v _ {, x} \\ \vdots \\ w _ {, z} \end{array} \right\}, \quad \text { where } \quad [ \mathbf {H} ] _ {6 \times 9} = \left[ \begin{array}{c c c c c c c c c} 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\ 0 & 1 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 1 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & 1 & 0 & 0 \end{array} \right] \tag {6.7-5}

Shape functions of a linear solid element (also called an eight-node brick element), Fig. 6.7-1, are

N _ {i} = \frac {1}{8} (1 \pm \xi) (1 \pm \eta) (1 \pm \zeta) \tag {6.7-6}

in which i = 1, 2, \ldots, 8 and the choice of algebraic signs should be obvious to an adequately prepared reader.

A quadratic “serendipity” solid element has corner nodes and a node on each edge, for a total of 20 nodes. A quadratic “Lagrange” solid element also includes midface nodes and a node at \xi = \eta = \zeta = 0 , for a total of 27 nodes. Shape functions of these quadratic elements are analogous to those in Eqs. 6.6-1 and Table 6.6-1 and may be found in [2.1]. Stiffness matrices of these elements are, respectively, 60 by 60 and 81 by 81. Quadratic elements may be good enough to permit the structure of Fig. 6.7-2 to be modeled by a single layer of elements, except near the fillet where stress gradients may be large.

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y, v z, w x, u

Figure 6.7-2. Model of one octant of a cylinder-to-cylinder intersection.

6.8 TRIANGULAR ISOPARAMETRIC ELEMENTS

In Chapter 5, we discussed triangular elements that are restricted to have straight sides and evenly spaced side nodes (Fig. 6.8-1a). In the present section these restrictions are removed, to allow triangles such as that in Fig. 6.8-1b, where sides need not be straight and side nodes need not be at midsides. We will see that the formulation procedure is essentially the same as that used for quadrilateral isoparametric elements.

Let \phi be a scalar field, interpolated from nodal values \phi_{i} . Similarly, x and y are coordinates interpolated from nodal values of x_{i} and y_{i} :

\phi = \sum N _ {i} \phi_ {i} \quad x = \sum N _ {i} x _ {i} \quad y = \sum N _ {i} y _ {i} \tag {6.8-1}

The element is isoparametric if the same shape functions N_{i} are used in all three summations. For a triangle, the N_{i} are expressed in terms of area coordinates \xi_{1}, \xi_{2} , and \xi_{3} , as explained in Section 5.3.

Imagine that the element characteristic matrix [\mathbf{k}] for a scalar field problem is required. Equations 6.6-4 through 6.6-7 may be used exactly as written, without even a change in symbols. We argue as follows.

Area coordinates are not independent. They satisfy the constraint relation \xi_1 + \xi_2 + \xi_3 = 1 . Accordingly, only two need be given to uniquely locate a point. For example, if \xi_1 and \xi_2 are given and the constraint relation is invoked, Eqs. 6.8-1 can be evaluated. We therefore define

\xi_ {1} = \xi

\xi_ {2} = \eta \tag {6.8-2}

\xi_ {3} = 1 - \xi - \eta

To evaluate the shape function derivatives seen in Eq. 6.6-5, we invoke the chain rule:

\frac {\partial N _ {i}}{\partial \xi} = \frac {\partial N _ {i}}{\partial \xi_ {1}} \frac {\partial \xi_ {1}}{\partial \xi} + \frac {\partial N _ {i}}{\partial \xi_ {2}} \frac {\partial \xi_ {2}}{\partial \xi} + \frac {\partial N _ {i}}{\partial \xi_ {3}} \frac {\partial \xi_ {3}}{\partial \xi} = \frac {\partial N _ {i}}{\partial \xi_ {1}} - \frac {\partial N _ {i}}{\partial \xi_ {3}} \tag {6.8-3}

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3 5 2 A₁ A₂ A₃ 6 4 1 y,v x,u

(a)

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3 5 2 6 4 1 y,v x,u

(b)

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B D M C C A D C D B M B D

(c)
Figure 6.8-1. (a) Quadratic triangle with straight sides and midside nodes, showing subareas A_{i} that define area coordinates \xi_{i} . (b) Quadratic triangle of arbitrary shape. (c) Sampling points used in quadrature formulas of Table 6.8-1.

Similarly

\frac {\partial N _ {i}}{\partial \eta} = \frac {\partial N _ {i}}{\partial \xi_ {2}} - \frac {\partial N _ {i}}{\partial \xi_ {3}} \tag {6.8-4}

For example, using the quadratic shape functions of Eqs. 5.3-5, we obtain for Eq. 6.6-5 the expression

\left[ \begin{array}{c} \mathbf {D} _ {N} \\ 2 \times 6 \end{array} \right] = \left[ \begin{array}{c c c c c c} 4 \xi_ {1} - 1 & 0 & - 4 \xi_ {3} + 1 & 4 \xi_ {2} & - 4 \xi_ {2} & 4 \left(\xi_ {3} - \xi_ {1}\right) \\ 0 & 4 \xi_ {2} - 1 & - 4 \xi_ {3} + 1 & 4 \xi_ {1} & 4 \left(\xi_ {3} - \xi_ {2}\right) & - 4 \xi_ {1} \end{array} \right] \tag {6.8-5}

[J] is produced by letting [D_{N}] premultiply a 6 by 2 array of nodal coordinates (let n = 6 in Eq. 6.6-4). Hence, [B] = [J]^{-1}[D_{N}] , as in Eq. 6.6-7.

Equation 6.6-8 is a valid expression for [k] of an n-node isoparametric triangle if the limits are 0 to 1 on the first integral and 0 to 1 - \eta on the second. Alternatively, we can write the following form, which is more appropriate to subsequent numerical integration:

\underset {n \times n} {[ \mathbf {k} ]} = \int_ {A} \underset {n \times 2} {[ \mathbf {B} ] ^ {T}} k [ \mathbf {B} ] t d A \tag {6.8-6}

Here t is element thickness, A is the area of the triangle, and [B] is a function of area coordinates \xi_{1} , \xi_{2} , and \xi_{3} .

Numerical Integration for Triangles. Let \phi be a function of area coordinates \xi_{1} , \xi_{2} , and \xi_{3} . The quadrature rule is

\int_ {A} \phi d A = \frac {1}{2} \sum_ {i = 1} ^ {n} W _ {i} J _ {i} \phi_ {i} \tag {6.8-7}

in which \phi_{i} is the value of \phi at a specific point in the triangle, W_{i} is the weight appropriate to this point, and n is the number of sampling points used. In going from Eq. 6.8-6 to Eq. 6.8-7 we set dA = J d\xi d\eta . The factor of \frac{1}{2} appears in Eq. 6.8-7 because the area of a reference triangle in area coordinates is \frac{1}{2} . For an undistorted triangle of unit area in Cartesian coordinates, J = 2 throughout. Hence, since \Sigma W_{i} = 1 (see data in Table 6.8-1), we obtain \int dA = 1 when \phi = 1 in Eq. 6.8-7.

Equation 6.8-7 must be applied to each term in the integrand of Eq. 6.8-6.

Equation 6.8-7 is similar in form to Eq. 6.4-6, but in Eq. 6.8-7 weights are expressed directly rather than as the product of one-dimensional weights. Data appear in Table 6.8-1 [6.5]. For points of multiplicity 3 (points B, C, D, and M) there are three points that have the same weight W_{i} . Area coordinates are given for one of the three; the other two are given by cyclic permutation. For example, using the first three-point formula in Table 6.8-1, we evaluate function \phi at the three locations

\begin{array}{c c c} (1) & (2) & (3) \\ \xi_ {1} = \frac {2}{3} & \xi_ {2} = \frac {2}{3} & \xi_ {3} = \frac {2}{3} \\ \xi_ {2} = \xi_ {3} = \frac {1}{6} & \xi_ {3} = \xi_ {1} = \frac {1}{6} & \xi_ {1} = \xi_ {2} = \frac {1}{6} \end{array} \tag {6.8-8}

TABLE 6.8-1. GAUSS QUADRATURE FORMULAS FOR INTEGRATION. OVER A TRIANGLE ACCORDING TO EQ. 6.8-7. APPROXIMATE LOCATIONS OF ENTRIES IN THE "POINTS" COLUMN ARE SHOWN IN FIG. 6.8-1C. ns = POINTS OF MULTIPLICITY 6, NOT SHOWN IN FIG. 6.8-1C.

Points	Multiplicity	Area Coordinates $\xi_1, \xi_2, \xi_3$				Weights $W_i$
		1-point formula		degree of precision 1
A	1	0.33333 33333 33333	0.33333	33333 33333	0.33333 33333 33333	1.00000 00000 00000
		3-point formula		degree of precision 2
B	3	0.66666 66666 66667	0.16666	66666 66667	0.16666 66666 66667	0.33333 33333 33333
		3-point formula		degree of precision 2
M	3	0.50000 00000 00000	0.50000	00000 00000	0.00000 00000 00000	0.33333 33333 33333
		4-point formula		degree of precision 3
A	1	0.33333 33333 33333	0.33333	33333 33333	0.33333 33333 33333	-0.56250 00000 00000
B	3	0.60000 00000 00000	0.20000	00000 00000	0.20000 00000 00000	0.52083 33333 33333
		6-point formula		degree of precision 4
B	3	0.81684 75729 80459	0.09157	62135 09771	0.09157 62135 09771	0.10995 17436 55322
C	3	0.10810 30181 68070	0.44594	84909 15965	0.44594 84909 15965	0.22338 15896 78011
		7-point formula		degree of precision 5
A	1	0.33333 33333 33333	0.33333	33333 33333	0.33333 33333 33333	0.22500 00000 00000
B	3	0.79742 69853 53087	0.10128	65073 23456	0.10128 65073 23456	0.12593 91805 44827
C	3	0.47014 20641 05115	0.47014	20641 05115	0.05971 58717 89770	0.13239 41527 88506
		12-point formula		degree of precision 6
B	3	0.87382 19710 16996	0.06308	90144 91502	0.06308 90144 91502	0.05084 49063 70207
D	3	0.50142 65096 58179	0.24928	67451 70910	0.24928 67451 70910	0.11678 62757 26379
ns	6	0.63650 24991 21399	0.31035	24510 33784	0.05314 50498 44817	0.08285 10756 18374
		13-point formula		degree of precision 7
A	1	0.33333 33333 33333	0.33333	33333 33333	0.33333 33333 33333	-0.14957 00444 67682
D	3	0.47930 80678 41920	0.26034	59660 79040	0.26034 59660 79040	0.17561 52574 33208
B	3	0.86973 97941 95568	0.06513	01029 02216	0.06513 01029 02216	0.05334 72356 08838
ns	6	0.63844 41885 69810	0.31286	54960 04874	0.04869 03154 25316	0.07711 37608 90257

and apply weight W = \frac{1}{3} to each. Center point A has multiplicity 1. Points ns (not shown) have multiplicity 6. The formulas in Table 6.8-1 are symmetric in the area coordinates. These formulas were not derived by Gauss but are called Gaussian because sampling points are optimally placed rather than simply being located in a uniform pattern.

The “degree of precision” in Table 6.8-1 refers to the degree of the highest-order complete polynomial in Cartesian coordinates that is integrated exactly by a formula. If \phi in Eq. 6.8-7 is not a polynomial, numerical integration is not exact but becomes more accurate as more sampling points are used. Inexact integration is expected for elements such as that in Fig. 6.8-1b, because geometric distortion produces a [B] matrix whose terms are not polynomials but rather the ratio of two polynomials.

Remarks. Like triangles, tetrahedra can be distorted from regular shapes such as that in Fig. 5.6-1. Triangles and tetrahedra decline in accuracy as sides become curved or side nodes become unevenly spaced. There is evidence that the ten-node tetrahedron is more sensitive to distortion than the six-node triangle.

If elements have straight sides and evenly spaced side nodes, one has the option of writing explicit expressions for stiffness coefficients rather than using numerical integration. The amount of algebra required to explicitly formulate a stiffness matrix in area or volume coordinates can be large. Fortunately, symbol-processing programs are available, of which MACSYMA may be the best known. Operations on polynomials, such as multiplication, factoring, differentiation, and integration can be performed symbolically, with output in the form of Fortran statements if so desired. The potential for saving time and reducing errors is obvious [6.6] .

6.9 CONSISTENT ELEMENT NODAL LOADS \{r_{e}\}

Consistent element nodal loads caused by initial strains \{\epsilon_{0}\} , initial stresses \{\sigma_{0}\} , body forces \{F\} , and surface tractions \{\Phi\} are computed according to Eq. 4.1-6. This matter is thoroughly discussed in Section 4.3. Results presented there remain valid if sides are straight and elements are rectangular; it does not matter that the elements may now be called isoparametric. For curved sides and general element shapes, results will differ and will be affected by the amount of geometric distortion.

Loads \{r_{e}\} caused by \{\epsilon_{0}\} , \{\sigma_{0}\} , and \{F\} are given by Eq. 6.3-21 for plane quadrilateral elements having any number of nodes. For solid hexahedra a triple integral is used, in which tJ d\xi d\eta is replaced by J d\xi d\eta d\zeta . For triangular elements integration spans the triangle area A, tJ d\xi d\eta is replaced by t dA, and formulas discussed in Section 6.8 are used.

A result of interest for the quadratic triangle is easy to show. Let the triangle be of constant thickness and have straight sides and midside nodes, as in Fig. 6.8-1a, and let there be a uniform body force in (say) the +x direction. Taking shape functions from Eq. 5.3-5, and using Eq. 5.2-8 for integrations, we obtain the following x-direction nodal loads:

\int_ {A} \left\lfloor \mathbf {N} \right] _ {6 \times 1} ^ {T} F _ {x} t d A = \frac {F _ {x} A t}{3} \left\lfloor 0 0 0 1 1 1 \right] ^ {T} \tag {6.9-1}

Thus the entire force F_{x}At is apportioned equally to only the midside nodes, regardless of the shape of the triangle (so long as sides remain straight and side nodes remain at midsides). This division also applies to the load produced by a uniform pressure against area A of the triangle. (Such a pressure loading may act on one six-node face of a ten-node tetrahedron, or on a six-node triangle formulated as a plate bending element.) If element sides are curved, the functions to be integrated are not polynomials: then numerical integration is required in Eq. 6.9-1, and the resulting nodal loads are not the same as those in Eq. 6.9-1.

Procedures for evaluating consistent loads \{\mathbf{r}_e\} caused by tractions on curved edges and warped surfaces are discussed in Section 5.8 of the second edition of this book.

6.10 THE VALIDITY OF ISOPARAMETRIC ELEMENTS

The critical test of element validity is the patch test, discussed in Section 4.6. In the present section we will stop short of the patch test, but will argue that the isoparametric formulation endows elements with all characteristics needed for convergence as discussed in Section 4.5. Elements discussed in the present chapter do in fact pass the patch test.

Isoparametric elements are well suited to problems whose functional II contains first spatial derivatives (or whose governing differential equations contain second spatial derivatives). Examples include heat conduction, plane stress analysis, and stress analysis of solids. We wish to show that isoparametric elements (a) provide C^{0} continuity (interelement continuity of the primary field variable), and (b) contain a complete linear polynomial in Cartesian coordinates [6.7].

Continuity. Interelement continuity of a field variable \phi can be demonstrated with the aid of Fig. 6.10-1. Along the common edge ABC , adjacent elements display the same edge-tangent coordinate: \eta_1 = \eta_2 . In addition, along edge ABC , shape functions of the two elements are identical functions of \eta and operate on d.o.f. of nodes A, B , and C only. Thus, whether viewed from element 1 or from element 2, the field variable along ABC is the same function. This argument also shows that elements match geometrically, because coordinates of A, B , and C define a unique quadratic curve.

Completeness. Imagine that \phi = \phi(x,y,z) is a polynomial field. If element nodes are attached to this field, so that nodal d.o.f. are \phi_{i} = \phi(x_{i},y_{i},z_{i}) , does shape function interpolation from nodal \phi_{i} yield the original field \phi throughout the element? The answer is yes if \phi is linear and the element is isoparametric. Specifically, we propose to show that \phi within an isoparametric element is a complete linear polynomial—that is, that

\phi = a _ {1} + a _ {2} x + a _ {3} y + a _ {4} z \tag {6.10-1}

when nodal d.o.f. \phi_{i} are consistent with this field and \phi within the element is evaluated by the interpolation

\phi = \sum N _ {i} \phi_ {i} \tag {6.10-2}

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C η₁ ξ₁ ① B η₂ ξ₂ ② A

Figure 6.10-1. Adjacent elements cited in compatibility arguments.

To do so we first evaluate Eq. 6.10-1 at each node:

\phi_ {i} = a _ {1} + a _ {2} x _ {i} + a _ {3} y _ {i} + a _ {4} z _ {i} \tag {6.10-3}

Here x_{i}, y_{i} , and z_{i} are Cartesian coordinates of node i, and i ranges over all nodes of the element, however many nodes there may be. Equations 6.10-2 and 6.10-3 yield

\phi = a _ {1} \sum N _ {i} + a _ {2} \sum N _ {i} x _ {i} + a _ {3} \sum N _ {i} y _ {i} + a _ {4} \sum N _ {i} z _ {i} \tag {6.10-4}

But, if the element is isoparametric, coordinates are interpolated in the same way as \phi —that is,

x = \sum N _ {i} x _ {i} \quad y = \sum N _ {i} y _ {i} \quad z = \sum N _ {i} z _ {i} \tag {6.10-5}

Accordingly, if \Sigma N_{i}=1 , Eq. 6.10-4 reduces to Eq. 6.10-1, and the proposition is proved.

To show that \Sigma N_{i}=1 , we note that Eqs. 6.10-5 can be used regardless of the location of the origin of Cartesian coordinates with respect to the element. Imagine that there is a second system XYZ, translated a distance h along the x axis so that x=X+h . Hence

x = \sum N _ {i} x _ {i} = \sum N _ {i} X _ {i} + h \sum N _ {i} = X + h \sum N _ {i} = (x - h) + h \sum N _ {i} \tag {6.10-6}

from which we obtain h = h \sum N_i and therefore conclude that \sum N_i = 1 .

Furthermore, from \Sigma N_{i} = 1 we obtain

\sum N _ {i, \xi} = \sum N _ {i, \eta} = \sum N _ {i, \zeta} = 0 \tag {6.10-7}

Equations 6.10-7 can be helpful in checking derivations and coding. Area coordinates for a triangle contain a redundant coordinate, which must be eliminated by means of Eqs. 6.8-2 in order for Eqs. 6.10-7 to be valid.

The foregoing arguments show that isoparametric elements have properties necessary to the passing of a patch test. They say nothing about accuracy in a coarse mesh, convergence rate with mesh refinement, or how accuracy declines as element geometry is distorted from a compact and regular shape (as in going from Fig. 6.6-1a to Fig. 6.6-1b). As sides become curved and as side nodes become unevenly spaced, an element tends to lose its ability to represent quadratic and higher polynomials in xyz coordinates. For example, an element that contains a

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y L 4 7 3 8 6 5 1 2 x

Figure 6.10.2. Superparametric element, in which \phi = \phi(\phi_1, \phi_2, \phi_3, \phi_4) .

complete quadratic polynomial in \xi\eta\zeta coordinates may not contain a complete quadratic polynomial in xyz coordinates, depending on the type of distortion from a regular shape. The argument of Eqs. 6.10-1 to 6.10-5 shows only that in spite of geometric distortion, a complete linear polynomial in xyz coordinates will remain.

Subparametric and Superparametric Elements. Such elements are defined in Section 6.1. For example, with straight sides and midside nodes, the element of Fig. 6.6-1a is subparametric, as its shape is then defined by the coordinates of nodes 1 through 4. The field quantity \phi is still defined by the \phi_i of all eight nodes. For such an element, the foregoing completeness argument remains valid. Specifically, for the element of Fig. 6.6-1a, summations in Eqs. 6.10-4 and 6.10-5 run from 1 to 8, but in Eqs. 6.10-5 the interpolations reduce to those of the bilinear element, Eqs. 6.3-1 and 6.3-2, provided that x_5 = (x_1 + x_2)/2 , y_5 = (y_1 + y_2)/2 , and so on. Thus the linear field of Eq. 6.10-1 is present when geometry is defined by only the four corner nodes.

Superparametric elements are usually not valid (however, with certain restrictions, valid superparametric elements for beams, plates, and shells are possible). Consider Fig. 6.10-2. Let the shape be defined by all eight nodes, that is, in a more general way than is the field quantity \phi , which is defined by \phi_i at nodes 1, 2, 3, and 4 only. Imagine that \phi_1 = \phi_4 = 0 while \phi_2 = \phi_3 = c , a constant. Thus we expect the element to display the constant gradient \phi_{,x} = c / L . However, gradient calculation according to Eq. 6.6-7 makes \phi_{,x} a function of \xi rather than constant, owing to [J] in Eq. 6.6-4, in which x_6 < L . The superparametric element of Fig. 6.10-2 would fail a patch test.

6.11 APPROPRIATE ORDER OF QUADRATURE

For numerically integrated elements, we define “full integration” as a quadrature rule sufficient to provide the exact integrals of all terms k_{ij} in the element stiffness matrix if the element is undistorted (e.g., if a quadratic element has straight sides and midside nodes). The same “full integration” rule will not exactly integrate all k_{ij} if sides are curved or if side nodes are offset from the midpoints, for then J is not constant throughout the element.

For example, in Eq. 6.2-6, [B] is linear in \xi and J is constant if node 3 is centered. Therefore, the integrand contains terms up to \xi^2 , which are integrated exactly by two Gauss points. Accordingly, for this element, even if node 3 is not centered, two-point Gauss quadrature is considered "full integration."

Use of full integration is the only sure way to avoid pitfalls such as mesh instabilities, which are discussed in Section 6.12.

However, a lower-order quadrature rule, called “reduced integration,” may be desirable for two reasons. First, since the expense of generating a matrix [k] by numerical integration is proportional to the number of sampling points, using fewer sampling points means lower cost. Second, a low-order rule tends to soften an element, thus countering the overly stiff behavior associated with an assumed displacement field. Softening comes about because certain higher-order polynomial terms happen to vanish at Gauss points of a low-order rule, so that these terms make no contribution to strain energy. In other words, with fewer sampling points, some of the more complicated displacement modes offer less resistance to deformation. In sum, our argument is that reduced integration may be able to simultaneously reduce cost, reduce accuracy in the evaluation of integral expressions, and increase the accuracy of a finite element analysis. Reduced integration should not be used if cost reduction is the only motivation.

The number of Gauss points has a lower limit because in the limit of mesh refinement, element volume must be integrated exactly. We argue as follows. As a mesh is indefinitely refined, a constant-strain condition is approached in each element, provided that the element is valid in the patch test sense. Thus strain energy density U_{0} becomes constant throughout each element. Strain energy in an element, for plane and solid problems, respectively, is

U _ {e} = \int \int U _ {0} t J d \xi d \eta \quad \text { or } \quad U _ {e} = \int \int \int U _ {0} J d \xi d \eta d \zeta \tag {6.11-1}

If U_{0} is constant, then U_{e} will be correct if volume dV = tJ \, d\xi \, d\eta (or dV = J \, d\xi \, d\eta \, d\zeta ) is correctly integrated. In practice, we prefer to use exact volume integration for any shape and size of element.

From Eq. 4.1-10 we see that integrals in Eq. 6.11-1 produce terms in the element stiffness matrix [k]. Accordingly, if [k] is produced by an integration rule adequate to compute element volume exactly, the element will be able to provide the correct strain energy in a constant-strain deformation mode.

Thus, for an element of arbitrary geometry, the minimum quadrature requirement is a rule that exactly integrates tJ (plane case) or J (solid case). In a plane bilinear element of constant thickness, tJ is linear in \xi and in \eta , so one Gauss point is required. In a plane quadratic element of constant thickness, tJ contains \xi^{3} and \eta^{3} , so a 2 by 2 Gauss rule is required. The eight-node solid also requires an order 2 Gauss rule (8 points).

However, with rare practical exceptions, indefinitely repeated subdivision of a mesh yields elements that become straight-sided parallelograms of constant thickness. Thus t and J cease to be functions of the coordinates and, in the limit, a single Gauss point yields the correct element volume.

For an isoparametric element based on an assumed displacement field, the best quadrature rule is usually the lowest-order rule that computes volume correctly and does not produce instability. Numerical testing of any proposed rule is mandatory. Solution accuracy may be mesh-dependent and problem-dependent, but usually one quadrature rule will be clearly superior to others. For bilinear and eight-node plane elements, and for the eight-node linear solid element, an order 2 Gauss rule is favored (four and eight points for plane and solid elements, respectively). The quadratic serendipity solid, having eight corner nodes and twelve edge nodes, can be integrated with an order 3 rule (27 points), but a special 14-point rule may be preferred, especially if the element is made very thin in one direction [6.8-6.10].

6.12 ELEMENT AND MESH INSTABILITIES

An instability may also be called a spurious singular mode. In structural mechanics, an instability may be known as a mechanism, a kinematic mode, an hourglass mode, or a zero-energy mode. The term “zero-energy mode” refers to a nodal displacement vector \{D\} that is not a rigid-body motion but nevertheless produces zero strain energy \{D\}^{T}[K]\{D\}/2 . Instabilities arise because of shortcomings in the element formulation process, such as use of a low-order Gauss quadrature rule. In the present context, an instability has nothing to do with buckling problems of structures.

A structure that appears adequately constrained may yet have an upper limit that makes [K] singular. Or, unstable elements may combine to form a structure that is stable but unduly susceptible to certain load patterns, so that computed displacements are excessive.

To explain the term “zero-energy mode” further and show that the term may arise, we substitute the relation \{\epsilon\} = [B]\{d\} into the expression for strain energy in an element, U_{e} . From Eq. 4.1-10 and the standard expression for [k], Eq. 4.1-5, we obtain

U _ {e} = \frac {1}{2} \{\mathbf {d} \} ^ {T} [ \mathbf {k} ] \{\mathbf {d} \} = \frac {1}{2} \{\mathbf {d} \} ^ {T} \int_ {V _ {e}} [ \mathbf {B} ] ^ {T} [ \mathbf {E} ] [ \mathbf {B} ] d V \{\mathbf {d} \} = \frac {1}{2} \int_ {V _ {e}} \{\boldsymbol {\epsilon} \} ^ {T} [ \mathbf {E} ] \{\boldsymbol {\epsilon} \} d V \tag {6.12-1}

When [\mathbf{k}] is formed by numerical integration, it contains only the information that can be sensed at the sampling points of the quadrature rule. If it happens that strains \{\epsilon\} = [\mathbf{B}]\{\mathbf{d}\} are zero at all sampling points for a certain mode \{\mathbf{d}\} , then U_{e} will vanish for that \{\mathbf{d}\} , and, according to Eq. 6.12-1, [\mathbf{k}] will be a zero-stiffness matrix in the sense that strain energy U_{e} = \{\mathbf{d}\}^{T}[\mathbf{k}]\{\mathbf{d}\} /2 is zero for this particular \{\mathbf{d}\} . We expect that U_{e} = 0 if \{\mathbf{d}\} is a rigid-body motion. If U_{e} = 0 when \{\mathbf{d}\} is a rigid-body motion, then an instability is present.

not a rigid-body motion, then an instability is present. An element that displays a mechanism is said to be rank deficient. That is, the rank of [k] is less than the number of element d.o.f. minus the number of rigid-body modes.

An instability in an existing [k] can be detected by means of an eigenvalue test (Section 18.8). In the present section we give examples of instabilities and briefly discuss their prevention.

Examples. Consider the four-node plane (bilinear) element, whose stiffness matrix is 8 by 8. Eight independent displacement modes \{\mathbf{d}\} can be identified (Fig. 6.12-1). The first three are rigid-body modes, for which U_{e} = 0 , as is correct, regardless of the quadrature rule used. The next three modes, numbers 4, 5, and 6, are constant-strain modes, for which U_{e} > 0 , regardless of the quadrature rule used. Modes 7 and 8 are bending modes. An order 1 rule, whose single Gauss point is at the element center, does not sense these modes, as \epsilon_{x} = \epsilon_{y} = \gamma_{xy} = 0 at the center. Accordingly, U_{e} = 0 for modes 7 and 8, and the element displays two mechanisms. These two spurious modes disappear if the Gauss rule is order 2 or greater.

2 or greater. The foregoing mechanisms can appear in a mesh of elements as well as in a single element (Fig. 6.12-2). In Fig. 6.12-2d, modes 7 and 8 of Fig. 6.12-1 are combined with a rigid-body rotation of each element. The mechanisms of Fig.