Applied Mechanics of Solids - Theory and Implementation of the Finite Element Method

Chapter 8

Theory and Implementation of the Finite Element Method The derivation and implementation of the finite element method outlined in the previous chapter is simple andeasy to follow, but it gives the misleading impression that the finite element method relies on the principle ofminimum potential energy, and so is applicable only to linear elastic solids. This is not the case, of course thefinite element method can solve problems involving very complex materials and large shape changes. This chapter contains

A more rigorous derivation of the finite element equations, based on the principle of virtual work, whichwas derived in Section 2.4.

1.

A more sophisticated implementation of finite element method for static linear elasticity. This includesmore accurate element interpolation schemes, and also extends the finite element method to threedimensions.

2.

A discussion of time integration schemes that are used in finite element simulations of dynamic problems,and a discussion of modal techniques for dynamic linear elasticity problems;

3.

An extension of the finite element method to nonlinear materials, using the hypoelastic material modeldescribed in Section 3.2 as a representative nonlinear material;

4.

An extension of the finite element method to account for large shape changes, using finite strain elasticityas a representative example;

5.

A discussion of finite element procedures for history dependent solids, using small strain viscoplasticityas a representative example.

6.

A discussion of the phenomenon of `locking’ that can cause the standard finite element method to fail incertain circumstances. Several techniques for avoiding locking are presented.

7.

In addition, a set of sample finite element codes (implemented in MAPLE and MATLAB) are provided toillustrate the how the various finite element procedures are implemented in practice. 8.1 Generalized FEM for static linear elasticity This section gives a more general derivation and implementation of the finite element method for static linearelastic solids than the energy-based derivation given in Chapter 7. 8.1.1 Review of the principle of virtual work Governing equations: We begin by summarizing the usualgoverning equations of linear elasticity, which must besolved by the FEA code. Given:

1. The shape of the solid in its unloaded condition 2. The initial stress field in the solid (we will take this

to be zero in setting up our FEM code)3. The elastic constants for the solid 4. The thermal expansion coefficients for the solid, and temperature distribution (we will take this to be

zero for our FEM code, for simplicity)5. A body force distribution acting on the solid (Note that in this section we will use b to denote force

per unit volume rather than force per unit mass, to avoid having to write out the mass density all thetime)

6. Boundary conditions, specifying displacements on a portion or tractions on a portion ofthe boundary of R

Calculate displacements, strains and stresses satisfying the governing equations of static linearelasticity

1. The strain-displacement equation

2. The elastic stress-strain law

3. The equation of static equilibrium for stresses

4. The boundary conditions on displacement and stress

Applied Mechanics of Solids (A.F. Bower) Chapter 8: Theory of FEA -... http://solidmechanics.org/text/Chapter8_1/Chapter8_1.htm

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The principle of virtual work: As we discussed in section 2.4, the principle of virtual work can be used toreplace the stress equilibrium equations. To express the principle, we define a kinematically admissible virtual displacement field , satisfying

on . You can visualize this field as a small change in the displacement of the solid, if you like, but itis really just an arbitrary differentiable vector field. The term `kinematically admissible’ is just a complicatedway of saying that on - that is to say, if you perturb the displacement slightly, the boundaryconditions on displacement are still satisfied. In addition, we define an associated virtual strain field

The principle of virtual work states that if the stress field satisfies

for all possible virtual displacement fields and corresponding virtual strains, it will automatically satisfy theequation of stress equilibrium , and also the traction boundary condition 8.1.2 Integral (weak) form of the governing equations of linear elasticity The principle of virtual work can be used to write the governing equation for the displacement field in a linearelastic solid in an integral form (called the `weak form’). Instead of solving the governing equations listed in thepreceding section, the displacements, strains and stresses are calculated as follows.

1. Find a displacement field satisfying

for all virtual velocity fields satisfying on .

2. Compute the strains from the definition

3. Compute the stresses from the stress-strain law The stress will automatically satisfy the equilibrium equation and boundary conditions, so all the field equationsand boundary conditions will be satisfied. The significance of this result is that it replaces the derivatives in the partial differential equations ofequilibrium with an equivalent integral, which is easier to handle numerically. It is essentially equivalent toreplacing the equilibrium equation with the principle of minimum potential energy, but the procedure based onthe principle of virtual work is very easily extended to dynamic problems, other stress-strain laws, and even toproblems involving large shape changes. Derivation: start with the virtual work equation

Recall that , and that , so that

Finally, recall that and that the elastic compliances must satisfy

so that . Finally, this shows that

Substituting into the virtual work equation gives the result we need. 8.1.3 Interpolating the displacement field and the virtual velocity field To solve the integral form of the elasticity equations given in 8.1.2, wediscretize the displacement field. That is to say, we choose to calculate thedisplacement field at a set of n discrete points in the solid (called `nodes’ infinite element terminology). We will denote the coordinates of these specialpoints by , where the superscript a ranges from 1 to N. The unknown


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displacement vector at each nodal point will be denoted by .

The displacement field at an arbitrary point within the solid will be specified byinterpolating between nodal values in some convenient way. An efficient and robust implementation of the finiteelement method requires a careful choice of interpolation scheme, but for now we will denote the interpolation ina general way as

Here, x denotes the coordinates of an arbitrary point in the solid. The interpolation functions arefunctions of position only, which must have the property that

for all b=1…N. (This is to make sure that the displacement field has the correct value at each node). Recentlydeveloped meshless finite element methods use very complex interpolation functions, but the more traditionalapproach is to choose them so that

The simple constant strain triangle elements introduced in 7.1 are one example of this type of interpolationscheme. We will define more complicated interpolation functions shortly. We can obviously interpolate the virtual velocity field in exactly the same way (since the principle of virtualwork must be satisfied for all virtual velocities, it must certainly be satisfied for an interpolated velocityfield…) so that

where are arbitrary nodal values of virtual velocity. 8.1.4 Finite element equations Substituting the interpolated fields into the virtual work equation, we find that

where summation on a and b is implied, in addition to the usual summation on i,j,k,l.

Note that the interpolation functions are known functions of position. We can therefore re-write the virtual workequation in matrix form as

where

Here K is known as the `stiffness matrix’ and f is known as the force vector. K is a function only of the elasticproperties of the solid, its geometry, and the interpolation functions and nodal positions. It is therefore a knownmatrix. Similarly, f is a function only of the known boundary loading and body force field, and the interpolationscheme and nodal positions. Observe that the symmetry of the elasticity tensor implies that K also has somesymmetry specifically . The virtual work equation must be satisfied for all possible sets of with for nodes a that lie on

. At these nodes, the displacements must satisfy . Evidently, this requires

This is a system of n linear equations for the n nodal displacements. 8.1.5 Simple 1D Implementation of the finite element method Before describing a fully general 3D implementation of the finiteelement method, we will illustrate all the key ideas using a simple 1-Dexample. Consider a long linear elastic bar, as shown in the picture. Assume

1. The bar has shear modulus and Poisson’s ratio 2 Th b h i d l h L


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Linear 1-D element

Quadratic 1-D element

2. The bar has cross section and length L3. It is constrained on all its sides so that 4. The bar is subjected to body force ,5. The bar is either loaded or constrained at its ends, so that the boundary conditions are either

or displacement at x=0 and x=L. For this 1-D example, then, the finite element equations reduce to

where

We could obviously choose any interpolation scheme, evaluate the necessary integrals and solve the resultingsystem of equations to compute the solution. It turns out to be particularly convenient, however, to use apiecewise-Lagrangian interpolation scheme, and to evaluate the integrals numerically using a Gaussianquadrature scheme.

To implement the Lagrangian interpolation scheme, we sub-dividethe region into a series of elements, as illustrated in thefigure. Each element is bounded by two nodal points, and mayalso contain one or more interior nodes. The displacement fieldwithin the element is interpolated between the nodes attached tothe element. So, we would use a linear interpolation between thenodes on a 2-noded element, a quadratic interpolation between thenodes on a 3 noded element, and so on. Generic linear and quadrilateral 1-D elements areillustrated in the table. The local nodes on theelement are numbered 1 and 2 for the linearelement, and 1,2,3 for the quadratic element asshown. We suppose that the element lies in theregion . The displacements within theelement are then interpolated as

where denotes the number of nodes on the element, denotes the value of the displacement at each node,and the shape functions are given in the table. Of course, the actual nodal coordinates do not lie at 1, +1 and 0 for all the elements. For a general element, wemap this special one to the region of interest. A particularly convenient way to do this is to set

where denotes the coordinate of each node on the element, and is the number of nodes on the element (2or 3). Elements that interpolate displacements and position using the same shape functions are calledisoparametric elements. Next, we need to devise a way to do the integrals in the expressions for the stiffness matrix and force vector. We can evidently divide up the integral so as to integrate over each element in turn

where is the total number of elements, and and denote the coordinates of the ends of the lthelement. We now notice an attractive feature of our interpolation scheme. The integral over the lth elementdepends only on the shape functions associated with the nodes on the lth element, since the displacement in thisregion is completely determined by its values at these nodes. We can therefore define element stiffness matrix,and element force matrix


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1-D integration points and weights

M=1

M=2

M=3

for each element, which depend on the geometry, interpolation functions and material properties of the element. The first and last elements have additional contributions to the element force vector from the boundary terms

. The global stiffness matrix is computed by summing all the element stiffness matrices

Finally we need to devise a way to compute the integrals for each element stiffness matrix. It is convenient tomap the domain of integration to [-1,+1] and integrate with respect to the normalized coordinate - thus

where is the Jacobian associated with the mapping, which may be computed as

Note that the mapping also enables us to calculate theshape function derivatives in the element stiffnessmatrix as

Finally, note that integrals may be computednumerically using a quadrature formula, as follows

where I=1…M denotes a set of integrationpoints in the region [-1,+1], and is a set ofintegration weights, which are chosen so as to makethe approximation as accurate as possible. Values aregiven in the table to the right for and 3.Higher order integration schemes exist but are requiredonly for higher order elements. For the linear 1-D element described earlier a single integration point issufficient to evaluate the stiffness exactly. Similarly, for the quaratic element, two integration points will suffice. 8.1.6 Summary of the 1D finite element procedure To summarize, then, the finite element solution requires the following steps:

1. For each element, compute the element stiffness matrix as follows:

where

and the integration points are tabulated above, and shape functions

were listed earlier.2. Assemble the contribution from each element to the global stiffness

3. Similarly, if there is a non-zero body force, then compute for each element

and assemble the global force vector

4. Add contributions to the force vector from prescribed traction boundary conditions at and

h th i t d t th d th t li t L


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where the superscript denotes the node that lies at x=L.5. Modify the stiffness matrix to enforce the constraints

6. Solve the system of linear equations

for the unknown displacements

8.1.7 Example FEM Code and solution A simple example MAPLE code for this 1-D example can be found inthe file FEM_1D_Static.mws It is set up to solve for displacements for a bar with the followingparameters:

1. Length L=5, unit x-sect area,2. Shear modulus 50, Poisson’s ratio 0.3,3. Uniform body force magnitude 10,4. Displacement u=0 at x=05. Traction t=2 at x=L.

The code computes the (1D) displacement distribution in the bar. Thepredicted displacement field is plotted on the right.

Of course in general we want to calculate more than just displacements usually we want the stress field too. We can calculate the stress field

anywhere within an element by differentiating the displacements tocalculate strains, and then substituting into the constitutive relation. This gives

This works well for a uniform body force with quadratic (3 nodedelements) as the plot on the right shows.

However, if we switch to linear elements, the stress results are not sogood (displacements are still calculated exactly). In this case, thestress must be uniform in each element (because strains are constantfor linear displacement field), so the stress plot looks like the figure tothe right.

Notice that the stresses are most accurate near the center of eachelement (at the integration point). For this reason, FEM codesgenerally output stress and strain data at integration points. It is interesting also to examine the stiffness matrix (shown below for 3linear elements, before addition of the u=0 constraint for the first node)

Notice that stiffness is symmetric, as expected, and also banded. A large FEM matrix is sparse most of theelements are zero. This allows the matrix to be stored in compact form for very large matrices indexedstorage (where only the nonzero elements together with their indices are stored) is the best approach; for smallerproblems skyline storage or band storage (where only the central, mostly nonzero, band of the matrix is stored)may be preferable. In this case equation numbers need to be assigned to each degree of freedom so as tominimize the bandwidth of the stiffness matrix. 8.1.8 Extending the 1D finite element method to 2 and 3 dimensions It is straightforward to extend the 1-D case to more general problems. Allthe basic ideas remain the same. Specifically

1. In both 2D and 3D we divide up our solid of interest into a


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number of elements, shown schematically for a 2D region in thepicture on the right.

2. We define interpolation functions for each element interms of a local, dimensionless, coordinate system within theelement. The coordinates satisfy . The displacementfield and the position of a point inside an element are computed interms of the interpolation functions as

where denote the shape functions, denote the displacement values and coordinates of thenodes on the element, and is the number of nodes on the element.

3. We introduce an element stiffness matrix for each element by defining

where denotes the element stiffness matrix for the (lth) element, and denotes the volume (in

3D) or area (in 2D) of the (lth) element, while denotes the surface of the (lth) element4. The volume integrals over each element are calculated by expressing the volume or surface integral in

terms of the dimensionless coordinates , and then evaluating the integrals numerically, usinga quadrature formula of the form

Here, are a set of integration weights (just numbers), and are a set of coordinates thatare selected to make the integration scheme as accurate as possible (also just numbers).

5. The global stiffness matrix

is then computed by summing the contribution from each element as

6. The stiffness matrix is modified to enforce any prescribed displacements7. The system of equations

is solved for the unknown nodal displacements.8. The stresses and strains within each element are then deduced.

To implement this procedure, we must (a) Define the element interpolation functions; (b) Express the integralsfor the element stiffness matrices and force vectors in terms of normalized coordinates; (c) Formulate anumerical integration scheme to evaluate the element stiffness matrices and force vectors.These details are addressed in the sections to follow. 8.1.9 Interpolation functions for 2D elements The 2D interpolation functions listed below are defined for the region

The numbers shown inside the element show the convention used to number the element faces.

2D interpolation functions


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8.1.10 Interpolation Functions for 3D elements The 3D interpolation functions listed below are defined for the region

3D Interpolation Functions


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The element faces are numbered as follows.

Linear and quadratic tetrahedral

Face 1 has nodes 1,2,3Face 2 has nodes 1,4,2Face 3 has nodes 2,4,3Face 4 has nodes 3,4,1

Linear and quadratic brick elements

Face 1 has nodes 1,2,3,4Face 2 has nodes 5,8,7,6Face 3 has nodes 1,5,6,3Face 4 has nodes 2,6,7,3Face 5 has nodes 3,7,8,4Face 6 has nodes 4,8,5,1

8.1.11 Volume integrals for stiffness and force in terms of normalized coordinates In this section we outline the procedure that is used to re-write the integrals for the element stiffness and force interms of the normalized coordinates . The integrals are

To evaluate them, we need to1. Find a way to calculate the derivatives of the shape functions in terms of 2. Map the volume (or area) integral to the region

Calculating the shape function derivatives. The shape function derivatives can be evaluated by writing

where the derivatives are easy to compute (just differentiate the expressions given earlier…). Tocompute recall that the coordinates of a point at position within an element can be determined as

where denotes the number of nodes on the element. Therefore

Note that is a 2x2 matrix (in 2D) or a 3x3 matrix (in 3D). Finally, follows as the inverse of thismatrix

Mapping the volume integral: To map the region of integration we define

where the matrix was defined earlier. Then the integral with respect to x is mapped into an integral with

respect to by setting

We note in passing that the boundary integral in the element force vector can be regarded as a 1-D line integralfor 2D elements and a 2D surface integral for 3D elements. So the procedures we developed in 8.1.5 (1Delements) can be used to evaluate the surface integral for a 2D element. Similarly, the procedures we developto integrate stiffness matrices for 2D elements can be used to evaluate the surface integral for a 3D element. 8.1.12 Numerical integration schemes for 2D and 3D elements Finally, to evaluate the integrals, we once again adopt a quadrature scheme, so that


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The integration points and weights depend on the element geometry, and are listed below for a fewcommon element types

Integration points for triangular elements 1 point

3 point

or

(the first scheme here is optimal, but has some disadvantages forquadratic elements because the integration points coincide with themidside nodes. The second scheme is less accurate but more robust).

4 point

Integration points for tetrahedral elements

1 point

4 point

where

Quadrilateral and hexahedral elements

For quadrilateral elements we can simply regard the integral over 2 spatial dimensions as successive 1-Dintegrals

which gives rise to the following 2D quadrature scheme: Let and for I=1…M denote 1-D quadraturepoints and weights listed below. Then in 2D, an quadrature scheme can be generated as follows:

for J=1…M and K=1…M let Similarly, in 3D, we generate an scheme as:for J=1…M , K=1…M L=1…M let

M=1 M=2

M=3

Choosing the number of integration points: There are two considerations. If too many integration points areused, time is wasted without gaining any accuracy). If too few integration points are used, the stiffness matrixmay be singular, or else the rate of convergence to the exact solution with mesh refinement will be reduced. Thefollowing schemes will avoid both

Number of integration points for fully integrated elementsLinear triangle (3 nodes): 1 pointQuadratic triangle (6 nodes): 4 points

Linear tetrahedron (4 nodes): 1 pointQuadratic tetrahedron (10 nodes): 4 points


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No._material_props: 3 Shear_modulus: 10. Poissons_ratio: 0.3 Plane strain/stress: 1No._coords_per_node: 2No._DOF_per_node: 2No._nodes: 4Nodal_coords: 0.0 0.0 1.0 0.0 1.0 1.0 0.0 1.0No._elements: 2Max_no._nodes_on_any_one_element: 3element_identifier; no._nodes_on_element; connectivity: 1 3 1 2 4 1 3 2 3 4No._nodes_with_prescribed_DOFs: 3Node_#, DOF#, Value: 1 1 0.0

Linear quadrilateral (4 nodes): 4 pointsQuadratic quadrilateral (8 nodes): 9 points

Linear brick (8 nodes): 8 pointsQuadratic brick (20 nodes): 27 points

There are situations where it is preferable to use fewer integration points and purposely make the stiffnesssingular. These are discussed in more detail in Section 8.5. 8.1.13 Summary of formulas for element stiffness and force matrices With these definitions, then, we write the element stiffness matrix as

where

8.1.14 Sample 2D/3D linear elastostatic FEM code You can find a MAPLE implementation of a simple 2D/3D static linear elasticitycode in the file FEM_2Dor3D_linelast_standard.mws The code reads an input file. Several examples are provided :

1. Linear_elastic_triangles.txt: Simple 2D plane strain problem with twotriangular elements

2. Linear_elastic_quad4.txt: Simple 2D plane strain problem with eight 4noded quadrilateral elements

3. Linear_elastic_quad8.txt: Simple 2D plane strain problem with two 8 noded quadrilateral elements4. Linear_elastic_brick4.txt: Simple 3D problem with 8 noded brick elements

As an example, we show how to run the program withthe first input file. The file sets up the problemillustrated in the figure above. The elements are linearelastic plane strain with . The program input file is listed on the right. Here is abrief explanation of the data in the file

1. The first part of the input file specifiesmaterial properties. A number ‘1’ on the Planestrain/stress line indicates a plane strainanalysis; a number ‘0’ indicates plane stress.

2. The second part specifies properties andcoordinates of the nodes. For a 2D problemeach node has 2 coordinates and 2 DOF; for a3D problem each node has 3 coordinates and3DOF. Then enter nodal coordinates for eachnode.

3. The third part lists the element properties. Here, you must specify the number of elements, and the maximum number of nodes on any one element(you can mix element types if you like). Then you must specify the nodes connected to each element(known as element connectivity). For each element, you must specify the number of nodes attached tothe element; an identifier that specifies the element type (you can enter any number in this version of thecode the identifier is provided to allow addition of more sophisticated element types such as reducedintegration elements), then enter the nodes on each element following the convention shown earlier.

4. The fourth part of the file specifies boundary constraints. For any constrained displacements, enter thenode number, the displacement component to be prescribed, and its value.

5. The last part of the file specifies distributed loading acting on the element faces. The loading isassumed to be uniform. For each loaded boundary, you should specify the element number, the face ofthe element (the face numbering convention was described in section 7.2.9 and 7.2.10 note that youmust be consistent in numbering nodes and faces on each element), and the components of tractionacting on the element face, as a vector with 2 or 3 components.

Note that the program performs absolutely no error checking on the input file. If you put in a typo, you will get

bi f A f d i l iff bl


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some bizzarre error message from MAPLE often during element stiffness assembly. For the input file shown, the program produces an output file that looks like this

The code prints the displacements at each node in the mesh, and also the strains and stresses at each integrationpoint (where these quantities are most accurate) for each element. To run the code, you must complete the following steps

1. Open the maple executable file;1. Edit the code to insert the full path for the input file in the line near the top of the code that reads

> # Change the name of the file below to point to your input file> infile :=fopen(`D:/fullpathoffile/Linear_elastic_triangles.txt`,READ):

2. Scroll down near the bottom to the line that reads> # Print nodal displacements, element strains and stresses to a file> #> outfile := fopen(`path/Linear_elastic_triangles.out`,WRITE):

and enter a name for the output file.3. Return to the top of the file, and press <enter> to execute each MAPLE block. You will see

the code plot the undeformed and deformed finite element mesh at the end. The stresses andstrains in the elements are printed to the output file.

(c) A.F. Bower, 2008This site is made freely available for educational purposes.

You may extract parts of the text for non-commercial purposes provided that the source is cited.Please respect the authors copyright.


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Applied Mechanics of Solids - Theory and Implementation of the Finite Element Method

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