310 Mechanics of Materials 2 $9.4 (d)Interior angles which are too extreme Interior angles which are excessively small or large will,like displaced"midside"nodes, cause distortion in the mapping process.A re-entrant corner (i.e.an interior angle greater than 180),see Fig.9.8,will cause failure in the mapping as the Jacobian matrix (relating the derivatives with respect to curvilinear(r,s),coordinates,to those with respect to carte- sian (x,y),coordinates),will not have an inverse (i.e.its determinate will be zero).For quadrilateral elements the ideal interior angle is 90,and for triangular elements it is 60. >180 c450 Re-entrant comer Fig.9.8.Extreme interior angles. (e)Warping Warping refers to the deviation of the face of a "planar"element from being planar,see Fig.9.9.The analogy of the three-legged milking stool (which is steady no-matter how uneven the surface is on which it is placed),to the triangular element serves to illustrate an advantage of this element over its quadrilateral counterpart. Non-planner element Fig.9.9.Warping
3 10 Mechanics of Materials 2 $9.4 (d) Interior angles which are too extreme Interior angles which are excessively small or large will, like displaced “midside” nodes, cause distortion in the mapping process. A re-entrant corner (i.e. an interior angle greater than 180”), see Fig. 9.8, will cause failure in the mapping as the Jacobian matrix (relating the derivatives with respect to curvilinear (rp), coordinates, to those with respect to cartesian (x,y), coordinates), will not have an inverse (i.e. its determinate will be zero). For quadrilateral elements the ideal interior angle is 90”, and for triangular elements it is 60”. Fig. 9.8. Extreme interior angles. (e) Warping Warping refers to the deviation of the face of a “planar” element from being planar, see Fig. 9.9. The analogy of the three-legged milking stool (which is steady no-matter how uneven the surface is on which it is placed), to the triangular element serves to illustrate an advantage of this element over its quadrilateral counterpart. Non-pbnner element Fig. 9.9. Warping
9.4 Introduction to the Finite Element Method 311 (f)Distortion Distortion is the deviation of an element from its ideal shape,which corresponds to that in curvilinear coordinates.SDRC I-DEAS13 gives two measures,namely (1)the departure from the basic element shape which is known as distortion,see Fig.9.10. Ideally,for a quadrilateral element,with regard to distortion,the shape should be a rectangle,and Fig.9.10.Distortion. (2)the amount of elongation suffered by an element which is known as stretch,or aspect ratio distortion,see Fig.9.11.Ideally,for a quadrilateral element,with regard to stretch, the shape should be square. Fig.9.11.Stretch. Whilst small amounts of deviation of an element's shape from that of the parent element can,and must,be tolerated,unnecessary and excessive distortions and stretch,etc.must be avoided if degraded results are to be minimised.High order elements in gradually varying strain fields are most tolerant of shape deviation,whilst low order elements in severe strain fields are least tolerant.>There are automatic means by which element shape deviation can be measured,using information derived from the Jacobian matrix.Errors in a solution and the rate of convergence can be judged by computing so-called energy norms derived from successive solutions.?However,it is left to the judgement of the user to establish the degree of shape deviation which can be tolerated.Most packages offer quality checking facilities, which allows the user to interrogate the shape deviation of all,or a selection of,elements.I- DEAS provides a measure of element quality using a value with a range of-1 to +1,(where
$9.4 Introduction to the Finite Element Method 31 1 If) Distortion Distortion is the deviation of an element from its ideal shape, which corresponds to that in curvilinear coordinates. SDRC I-DEASI3 gives two measures, namely (1) the departure from the basic element shape which is known as distortion, see Fig. 9.10. Ideally, for a quadrilateral element, with regard to distortion, the shape should be a rectangle, and Fig. 9.10. Distortion (2) the amount of elongation suffered by an element which is known as stretch, or aspect ratio distortion, see Fig. 9.1 1. Ideally, for a quadrilateral element, with regard to stretch, the shape should be square. Fig. 9.1 I. Stretch. Whilst small amounts of deviation of an element’s shape from that of the parent element can, and must, be tolerated, unnecessary and excessive distortions and stretch, etc. must be avoided if degraded results are to be minimised. High order elements in gradually varying strain fields are most tolerant of shape deviation, whilst low order elements in severe strain fields are least tolerant? There are automatic means by which element shape deviation can be measured, using information derived from the Jacobian matrix. Errors in a solution and the rate of convergence can be judged by computing so-called energy norms derived from successive solutions.’ However, it is left to the judgement of the user to establish the degree of shape deviation which can be tolerated. Most packages offer quality checking facilities, which allows the user to interrogate the shape deviation of all, or a selection of, elements. IDEAS provides a measure of element quality using a value with a range of -1 to +I, (where
312 Mechanics of Materials 2 $9.4 +I is the target value corresponding to zero distortion,and stretch,etc.).Negative values, which arise for example,from re-entrant corners,referred to above,will cause an attempted solution to fail,and hence need to be rectified.A distortion value above 0.7 can be considered acceptable,but errors will be incurred with any value below 1.0.However,circumstances may dictate acceptance of elements with a distortion value below 0.7.Similarly,as a rule- of-thumb,a stretch value above 0.5 can be considered acceptable,but again,errors will be incurred with any value below 1.0.Companies responsible for analyses may issue guidelines for quality,an example of which is shown in Table 9.1. Table 9.1.Example of element quality guidelines. Element Interior angle Warpage Distortion Stretch Triangle 30-90 N/A 0.35 0.3 Ouadrilateral 45-135 0.2 0.60 0.3 Wedge 30-90 N/A 0.35 0.3 Tetrahedron 30-90 N/A 0.10 0.1 Hexahedron 45-135 0.2 0.5 0.3 9.4.5.Creation of the material model The least material data required for a stress analysis is the empirical elastic modulus for the component under analysis describing the relevant stress/strain law.For a dynamic analysis, the material density must also be specified.Dependent upon the type of analysis,other properties may be required,including Poisson's ratio for two-and three-dimensional models and the coefficient of thermal expansion for thermal analyses.For analyses involving non- linear material behaviour then,as a minimum,the yield stress and yield criterion,e.g.von Mises,need to be defined.If the material within an element can be assumed to be isotropic and homogeneous,then there will be only one value of each material property.For non- isotropic material,i.e.orthotropic or anisotropic,then the material properties are direction and spacially dependent,respectively.In the extreme case of anisotropy,21 independent values are required to define the material matrix.5 9.4.6.Node and element ordering Before moving on to consider boundary conditions,it is appropriate to examine node and element ordering and its effect on efficiency of solution by briefly exploring the methods used.The formation of the element characteristic matrices (to be considered in $9.7,9.8 and 9.9),and the subsequent solution are the two most computationally intensive steps in any fe.analysis.The computational effort and memory requirements of the solution are affected by the method employed,and are considered below. It will be seen in Section 9.7,and subsequently,that the displacement based method involves the assembly of the structural,or assembled,stiffness matrix [K],and the load and displacement column matrices,(P)and (p),respectively,to form the governing equation for stress analysis (P)=[K](p).With reference to $9.7,and subsequently,two features of the fem.will be seen to be that the assembled stiffness matrix [K],is sparsely populated and is symmetric.Advantage can be taken of this in reducing the storage requirements of the
312 Mechanics of Materials 2 99.4 +I is the target value corresponding to zero distortion, and stretch, etc.). Negative values, which arise for example, from re-entrant comers, referred to above, will cause an attempted solution to fail, and hence need to be rectified. A distortion value above 0.7 can be considered acceptable, but errors will be incurred with any value below 1.0. However, circumstances may dictate acceptance of elements with a distortion value below 0.7. Similarly, as a ruleof-thumb, a stretch value above 0.5 can be considered acceptable, but again, errors will be incurred with any value below 1 .O. Companies responsible for analyses may issue guidelines for quality, an example of which is shown in Table 9.1. Table 9.1. Example of element quality guidelines. Element Interior angle" Warpage Distortion Stretch Triangle 30 - 90 N/A 0.35 0.3 Quadrilateral 45-135 0.2 0.60 0.3 Wedge 30-90 N/A 0.35 0.3 Tetrahedron 30-90 N/A 0.10 0.1 Hexahedron 45- 135 0.2 0.5 0.3 9.4.5. Creation of the material model The least material data required for a stress analysis is the empirical elastic modulus for the component under analysis describing the relevant stresshtrain law. For a dynamic analysis, the material density must also be specified. Dependent upon the type of analysis, other properties may be required, including Poisson's ratio for two- and three-dimensional models and the coefficient of thermal expansion for thermal analyses. For analyses involving nonlinear material behaviour then, as a minimum, the yield stress and yield criterion, e.g. von Mises, need to be defined. If the material within an element can be assumed to be isotropic and homogeneous, then there will be only one value of each material property. For nonisotropic material, i.e. orthotropic or anisotropic, then the material properties are direction and spacially dependent, respectively. In the extreme case of anisotropy, 21 independent values are required to define the material matrix? 9.4.6. Node and element ordering Before moving on to consider boundary conditions, it is appropriate to examine node and element ordering and its effect on efficiency of solution by briefly exploring the methods used. The formation of the element characteristic matrices (to be considered in §9.7,9.8 and 9.9), and the subsequent solution are the two most computationally intensive steps in any fe. analysis. The computational effort and memory requirements of the solution are affected by the method employed, and are considered below. It will be seen in Section 9.7, and subsequently, that the displacement based method involves the assembly of the structural, or assembled, stiffness matrix [K], and the load and displacement column matrices, {PI and {p), respectively, to form the governing equation for stress analysis (P) = [K]{p]. With reference to $9.7, and subsequently, two features of the fem. will be seen to be that the assembled stiffness matrix [K], is sparsely populated and is symmetric. Advantage can be taken of this in reducing the storage requirements of the
s94 Introduction to the Finite Element Method 313 computer.Two solution methods are used,namely,banded or frontal,the choice of which is dependent upon the number of dof.in the model. Banded method of solution The banded method is appropriate for small to medium size jobs(i.e.up to 10 000 dof.). By carefully ordering the dof.the assembled stiffness matrix [K],can be banded with non- zero terms occurring only on the leading diagonal.Symmetry permits only half of the band to be stored,but storage requirements can still be high.It is advantageous therefore to minimise the bandwidth.A comparison of different node numbering schemes is provided by Figs.9.12 and 9.13 in which a simple model comprising eight triangular linear elements is considered, and for further simplicity the nodal contributions are denoted as shaded squares,the empty squares denoting zeros. The semi-bandwidth can be seen to depend on the node numbering scheme and the number of dof.per node and has a direct effect on the storage requirements and computational effort. 1.2 3.4 5.6 7.8 9.10 2 Node number a e 目ement label 8 9 10 11.12 13.14 15.16 17.18 19@ Degree of froedom number (a)Poor node numbering scheme Semi-bandwidth =14 1234567891011121314151617181920 2 3 4 6 Contains non-zero terms ▣ Zero terms 890121345 1 1819 20 (b)Structural stiffess matrix with non-zero terms widely dispersed Fig.9.12.Structural stiffness matrix corresponding to poor node ordering
§9.4 Introduction to the Finite Element Method 313 computer. Two solution methods are used, namely, banded or frontal, the choice of which is dependent upon the number of dof. in the model. Banded method of solution The banded method is appropriate for small to medium size jobs (i.e. up to 10 000 dof.). By carefully ordering the dof. the assembled stiffness matrix [K], can be banded with nonzero terms occurring only on the leading diagonal. Symmetry permits only half of the band to be stored, but storage requirements can still be high. It is advantageous therefore to minimise the bandwidth. A comparison of different node numbering schemes is provided by Figs. 9.12 and 9.13 in which a simple model comprising eight triangular linear elements is considered, and for further simplicity the nodal contributions are denoted as shaded squares, the empty squares denoting zeros. The semi-bandwidth can be seen to depend on the node numbering scheme and the number of dof. per node and has a direct effect on the storage requirements and computational effort. Fig. 9.12. Structural stiffness matrix corresponding to poor node ordering
314 Mechanics of Materials 2 s9.4 For a given number of dof.per node,which is generally fixed for each assemblage,the bandwidth can be minimised by using a proper node numbering scheme. With reference to Figs.9.12 and 9.13 there are a total of 20 dof.in the model (i.e.10 nodes each with an assumed 2 dof.),and if the symmetry and bandedness is not taken advantage of, storage of the entire matrix would require 202=400 locations.For the efficiently numbered model with a semi-bandwidth of 8,see Fig.9.13,taking advantage of the symmetry and bandedness,the storage required for the upper,or lower,half-band is only 8x 20=160 locations. 1.2 5.6 9.10 13.14 17.18 3 5 7 (⑨ Node number d 6 Element label 2 4 6 10 3.4 7.8 11.12 15.16 1920 Degree of freedom number (a)Efficient node numbering scheme Semi-bandwidth =B 1 23456 9 1011121314151617181920 2 3 4 6 7 Contains non-zero temms 8 9 Z●ro terms 10 11 14 16 17 18 19 20 (b)Structural stiffness matrix with non-zero terms closely banded Fig.9.13.Structural stiffness matrix corresponding to efficient node ordering. From observation of Figs.9.12 and 9.13 it can be deduced that the semi-bandwidth can be calculated from semi-bandwidth f(d+1)
314 Mechanics of Materials 2 §9.4 For a given number of dof. per node, which is generally fixed for each assemblage, the bandwidth can be minimised by using a proper node numbering scheme. With reference to Figs. 9.12 and 9.13 there are a total of 20 dof. in the model (i.e. 10 nodes each with an assumed 2 dof.), and if the symmetry and bandedness is not taken advantage of, storage of the entire matrix would require 202 = 400 locations. For the efficiently numbered model with a semi-bandwidth of 8, see Fig. 9.13, taking advantage of the symmetry and bandedness, the storage required for the upper, or lower, half-band is only 8 x 20 = 160 locations. Fig. 9.13. Structural stiffness matrix corresponding to efficient node ordering. From observation of Figs. 9.12 and 9.13 it can be deduced that the semi-bandwidth can be calculated from semi-bandwidth = f(d + I)