§9.4 Introduction to the Finite Element Method 305 9.4.2.Identification of the type of analysis The most appropriate type(s)of analysis to be employed needs to be identified in order that the component behaviour can best be represented.The assumption of either plane stress or plane strain is a common example.The high cost of a full three-dimensional analysis can be avoided if the assumption of both geometric and load symmetry can be made.If the application calls for elastic stress analysis,then the system equations will be linear and can be solved by a variety of methods,Gaussian elimination,Choleski factorisation or Gauss-Seidel procedure.s For large displacement or post-yield material behaviour applications the system equations will be non-linear and iterative solution methods are required,such as that of Newton- Raphson.5 9.4.3.Idealisation Commercially available finite element packages usually have a number of different elements available in the element library.For example,one such package,HKS ABAQUS12 has nearly 400 different element variations.Examples of some of the commonly used elements have been given in Fig.9.1. Often the type of element to be employed will be evident from the physical application.For example,rod and beam elements can represent the behaviour of frames,whilst shell elements may be most appropriate for modelling a pressure vessel.Some regions which are actually three-dimensional can be described by only one or two independent coordinates,e.g.pistons, valves and nozzles,etc.Such regions can be idealised by using axisymmetric elements. Curved boundaries are best represented by elements having mid-side(or intermediate)nodes in addition to their corner nodes.Such elements are of higher order than linear elements (which can only represent straight boundaries)and include quadratic and cubic elements. The most popular elements belong to the so-called isoparametric family of elements,where the same parameters are used to define the geometry as define the displacement variation over the element.Therefore,those isoparametric elements of quadratic order,and above,are capable of representing curved sides and surfaces. In situations where the type of elements to be used may not be apparent,the choice could be based on such considerations as (a)number of dof., (b)accuracy required, (c)computational effort, (d)the degree to which the physical structure needs to be modelled. Use of the elements with a quadratic displacement assumption are generally recommended as the best compromise between the relatively low cost but inferior performance of linear elements and the high cost but superior performance of cubic elements. 9.4.4.Discretisation of the solution region This step is equivalent to replacing the actual structure or continuum having an infinite number of dof.by a system having a finite number of dof.This process,known as
$9.4 Introduction to the Finite Element Method 305 9.4.2. IdentiJication of the type of analysis The most appropriate type(s) of analysis to be employed needs to be identified in order that the component behaviour can best be represented. The assumption of either plane stress or plane strain is a common example. The high cost of a full three-dimensional analysis can be avoided if the assumption of both geometric and load symmetry can be made. If the application calls for elastic stress analysis, then the system equations will be linear and can be solved by a variety of methods, Gaussian elimination, Choleski factorisation or Gauss-Seidel procedure ? For large displacement or post-yield material behaviour applications the system equations will be non-linear and iterative solution methods are required, such as that of NewtonRaphson? 9.4.3. Idealisation Commercially available finite element packages usually have a number of different elements available in the element library. For example, one such package, HKS ABAQUS'* has nearly 400 different element variations. Examples of some of the commonly used elements have been given in Fig. 9.1. Often the type of element to be employed will be evident from the physical application. For example, rod and beam elements can represent the behaviour of frames, whilst shell elements may be most appropriate for modelling a pressure vessel. Some regions which are actually three-dimensional can be described by only one or two independent coordinates, e.g. pistons, valves and nozzles, etc. Such regions can be idealised by using axisymmetric elements. Curved boundaries are best represented by elements having mid-side (or intermediate) nodes in addition to their comer nodes. Such elements are of higher order than linear elements (which can only represent straight boundaries) and include quadratic and cubic elements. The most popular elements belong to the so-called isoparametric family of elements, where the same parameters are used to define the geometry as define the displacement variation over the element. Therefore, those isoparametric elements of quadratic order, and above, are capable of representing curved sides and surfaces. In situations where the type of elements to be used may not be apparent, the choice could be based on such considerations as (a) number of dof., (b) accuracy required, (c) computational effort, (d) the degree to which the physical structure needs to be modelled. Use of the elements with a quadratic displacement assumption are generally recommended as the best compromise between the relatively low cost but inferior performance of linear elements and the high cost but superior performance of cubic elements. 9.4.4. Discretisation of the solution region This step is equivalent to replacing the actual structure or continuum having an infinite number of dof. by a system having a finite number of dof. This process, known as
306 Mechanics of Materials 2 9.4 discretisation,calls for engineering judgement in order to model the region as closely as necessary.Having selected the element type,discretisation requires careful attention to extent of the model (i.e.location of model boundaries),element size and grading,number of elements,and factors influencing the qualiry of the mesh,to achieve adequately accurate results consistent with avoiding excessive computational effort and expense.These aspects are briefly considered below. Extent of model Reference has already been made above to applications which are axisymmetric,or those which can be idealised as such.Generally,advantage should be taken of geometric and loading symmetry wherever it exists,whether it be plane or axial.Appropriate boundary conditions need to be imposed to ensure the reduced portion is representative of the whole. For example,in the analysis of a semi-infinite tension plate with a central circular hole, shown in Fig.9.3,only a quadrant need be modelled.However,in order that the quadrant is representative of the whole,respective v and u displacements must be prevented along the x and y direction symmetry axes,since there will be no such displacements in the full model/component. (a)Actual component (b)Idealisation using graded triangular elements 0 (c)Direct stress distribution in y direction across lateral symmetry axis Fig.9.3.Finite element analysis of a semi-infinite tension plate with a central circular hole,using triangular elements. Further,it is known that disturbances to stress distributions due to rapid changes in geometry or load concentrations are only local in effect.Saint-Venant's principle states that the effect of stress concentrations essentially disappear within relatively small distances (approximately
306 Mechanics of Materials 2 §9.4 discretisation, calls for engineering judgement in order to model the region as closely as necessary .Having selected the element type, discretisation requires careful attention to extent of the model (i.e. location of model boundaries), eleme~t size and grading, number of elements, and factors influencing the quality of the mesh, to achieve adequately accurate results consistent with avoiding excessive computational effort and expense. These aspects are briefly considered below. Extent of model Reference has already been made above to applications which are axisymmetric, or those which can be idealised as such. Generally, advantage should be taken of geometric and loading symmetry wherever it exists, whether it be plane or axial. Appropriate boundary conditions need to be imposed to ensure the reduced portion is representative of the whole. For example, in the analysis of a semi-infinite tension plate with a central circular hole, shown in Fig. 9.3, only a quadrant need be modelled. However, in order that the quadrant is representative of the whole, respective v and u displacements must be prevented along the x and y direction symmetry axes, since there will be no such displacements in the full t:nodel/component. Fig. 9.3. Finite element analysis of a semi-infinite tension plate with a central circular hole, using triangular elements. Further, it is known that disturbances to stress distributions due to rapid changes in geometry or load concentrations are only local in effect. Saint-Venant's principle states that the effect of stress concentrations essentially disappear within relatively small distances (approximately
$9.4 Introduction to the Finite Element Method 307 equal to the larger lateral dimension),from the position of the disturbance.Advantage can therefore be taken of this principle by reducing the necessary extent of a finite element model.A rule-of-thumb is that a model need only extend to one-and-a-half times the larger lateral dimension from a disturbance,see Fig.9.4. (a)Actual component (b)Boundary for finite element idealisation 3b Fig.9.4.Idealisation of a shouldered tension strip. Element size and grading The relative size of elements directly affects the quality of the solution.As the element size is reduced so the accuracy of solution can be expected to increase since there is better representation of the field variable,e.g.displacement,and/or better representation of the geometry.However,as the element size is reduced,so the number of elements increases with the accompanying penalty of increased computational effort.Needlessly small elements in regions with little variation in field variable or geometry will be wasteful.Equally,in regions where the stress variation is not of primary interest then a locally coarse mesh can be employed providing it is sufficiently far away from the region of interest and that it still provides an accurate stiffness representation.Therefore,element sizes should be graded in order to take account of anticipated stress/strain variations and geometry,and the results required.The example of stress analysis of a semi-infinite tension plate with a central circular hole,Fig.9.3,serves to illustrate how the size of the elements can be graded from small-size elements surrounding the hole (where both the stress/strain and geometry are varying the most),to become coarser with increasing distance from the hole. Number of elements The number of elements is related to the previous matter of element size and,for a given element type,the number of elements will determine the total number of dof.of the model, and combined with the relative size determines the mesh density.An increase in the number of elements can result in an improvement in the accuracy of the solution,but a limit will be reached beyond which any further increase in the number of elements will not significantly improve the accuracy.This matter of convergence of solution is clearly important,and with experience a near optimal mesh may be achievable.As an alternative to increasing the number of elements,improvements in the model can be obtained by increasing the element order. This alternative form of enrichment can be performed manually (by substituting elements)
99.4 Introduction to the Finite Element Method 307 equal to the larger lateral dimension), from the position of the disturbance. Advantage can therefore be taken of this principle by reducing the necessary extent of a finite element model. A rule-of-thumb is that a model need only extend to one-and-a-half times the larger lateral dimension from a disturbance, see Fig. 9.4. (a) Actual component P +q7;TEy- (b) Boundary for finite element idealition _p 2 ._.___.__.____ Fig. 9.4. ldealisation of a shouldered tension strip. Element size and grading The relative size of elements directly affects the quality of the solution. As the element size is reduced so the accuracy of solution can be expected to increase since there is better representation of the field variable, e.g. displacement, and/or better representation of the geometry. However, as the element size is reduced, so the number of elements increases with the accompanying penalty of increased computational effort. Needlessly small elements in regions with little variation in field variable or geometry will be wasteful. Equally, in regions where the stress variation is not of primary interest then a locally coarse mesh can be employed providing it is sufficiently far away from the region of interest and that it still provides an accurate stiffness representation. Therefore, element sizes should be graded in order to take account of anticipated stresshtrain variations and geometry, and the results required. The example of stress analysis of a semi-infinite tension plate with a central circular hole, Fig. 9.3, serves to illustrate how the size of the elements can be graded from small-size elements surrounding the hole (where both the stresdstrain and geometry are varying the most), to become coarser with increasing distance from the hole. Number of elements The number of elements is related to the previous matter of element size and, for a given element type, the number of elements will determine the total number of dof. of the model, and combined with the relative size determines the mesh density. An increase in the number of elements can result in an improvement in the accuracy of the solution, but a limit will be reached beyond which any further increase in the number of elements will not significantly improve the accuracy. This matter of convergence of solution is clearly important, and with experience a near optimal mesh may be achievable. As an alternative to increasing the number of elements, improvements in the model can be obtained by increasing the element order. This alternative form of enrichment can be performed manually (by substituting elements)
308 Mechanics of Materials 2 $9.4 or can be performed automatically,e.g.the commercial package RASNA has this capability. Clearly,any increase in the number of elements (or element order),and hence dof.,will require greater computational effort,will put greater demands on available computer memory and increase cost. Quality of the mesh The quality of the fe.predictions (e.g.of displacements,temperatures,strains or stresses), will clearly be affected by the performance of the model and its constituent elements.The factors which determine quality13 will now be explored briefly,namely (a)coincident elements, (b)free edges, (c)poorly positioned“midside”nodes, (d)interior angles which are too extreme, (e)warping,and (①distortion. (a)Coincident elements Coincident elements refer to two or more elements which are overlaid and share some of the nodes,see Fig.9.5.Such coincident elements should be deleted as part of cleaning-up of a mesh. Typical coincident nodes Fig.9.5.Coincident elements (b)Free edges A free edge should only exist as a model boundary.Neighbouring elements should share nodes along common inter-element boundaries.If they do not,then a free edge exists and will need correction,see Fig.9.6. (c)Poorly positioned "midside"nodes Displacing an element's"midside"node from its mid-position will cause distortion in the mapping process associated with high order elements;and in extreme cases can significantly degrade an element's performance.There are two aspects to "midside"node displacement, namely.the relative position between the corner nodes,and the node's offset from a straight
308 Mechanics of Materials 2 $9.4 or can be performed automatically, e.g. the commercial package RASNA has this capability. Clearly, any increase in the number of elements (or element order), and hence dof., will require greater computational effort, will put greater demands on available computer memory and increase cost. Quality of the mesh The quality of the fe. predictions (e.g. of displacements, temperatures, strains or stresses), will clearly be affected by the performance of the model and its constituent elements. The factors which determine quality13 will now be explored briefly, namely coincident elements, free edges, poorly positioned “midside” nodes, interior angles which are too extreme, warping, and distortion. Coincident elements Coincident elements refer to two or more elements which are overlaid and share some of the nodes, see Fig. 9.5. Such coincident elements should be deleted as part of cleaning-up of a mesh. coincident nodes Fig. 9.5. Coincident elements. (b) Free edges A free edge should only exist as a model boundary. Neighbouring elements should share nodes along common inter-element boundaries. If they do not, then a free edge exists and will need correction, see Fig. 9.6. (e) Poorly positioned “midside” nodes Displacing an element’s “midside” node from its mid-position will cause distortion in the mapping process associated with high order elements, and in extreme cases can significantly degrade an element’s performance. There are two aspects to “midside” node displacement, namely, the relative position between the corner nodes, and the node’s offset from a straight
$9.4 Introduction to the Finite Element Method 309 Interior free edges Fig.9.6.Free edges. line joining the corner nodes,see Fig.9.7.The midside node's relative position should ideally be 50%of the side length for a parabolic element and 33.3%for a cubic element.An example of the effect of displacement of the"midside"node to the 25%position,is reported for a parabolic element4 to result in a 15%error in the major stress prediction. Percent displacement =100 b/c Offset-a/c Fig.9.7.“Midside”node displacement
39.4 Introduction to the Finite Element Method 309 Interior Fig 9 6 Free edges line joining the corner nodes, see Fig. 9.7. The midside node’s relative position should ideally be 50% of the side length for a parabolic element and 33.3% for a cubic element. An example of the effect of displacement of the “midside” node to the 25% position, is reported for a parabolic elementI4 to result in a 15% error nn the major stress prediction. Percent displacement = 100 b/c Fig 9 7 “Midside” node displacement Offset = a/c