1 mathematical review - rice university

Copyright JE Akin. All rights reserved. of 24

1 Mathematical Review

1.1 Introduction

The physical problem to be solved, u (x, y, z), is formulated initially in a physical region with dimensional spatial coordinates, (x, y, z). The physical problem is typically stated as a differential equation, involving u and its derivatives with respect to x, y, and z, on the interior of a physical region in space. That differential equation has essential and non-essential boundary conditions applied in non-overlapping portions of the boundary of the region. The equivalent integral form must be subject to the same essential boundary conditions and must be integrated over the same physical region. To be able to match curvilinear physical shapes it is necessary to convert (map) a non-dimensional parametric space to (part of) the physical space. In other words, a change of variables is required to convert physical quantities in a physical space (or time, or space-time) to an integral of physical quantities in a non-dimensional space.

1.2 Terminology from differential equations

1.2.1 Definitions

An ordinary differential equation (ODE) has a dependent variable, say u, and an independent variable (here usually the spatial coordinate x, or time t). The dependent variable is to be found for all values of the independent variable within some specified region. The boundary conditions are applied at the two bounding end points of that region.

Let x be the independent variable and let u(x) be the dependent variable to be determined inside a specific range of x. The differential equation is said to be of order n if the n-th derivative of u with respect to (wrt) x is the highest derivative in that equation. A PDE of order n can be written as a set of n first order differential equations. Some authors utilize that approach when applying the least squares FEA. That change allows the inter-element continuity to be reduced to simply having the function continuous.

In this course, we will consider mainly even order equations n=2m in space and first or second order in time. A first-order ODE is said to be linear it can be written in the form

( ) ( )

or ( ) ( ) where ( ) ⁄ , and f(x) and p(x) are any given functions of x. When p(x)=0, the equation is said to be homogeneous; otherwise it is call non-homogeneous. Likewise, a second-order ODE is said to be linear if it can be written as:

( )

( ) ( )

or u’’+f(x)u’+g(x)u=p(x). For example, u’’+2u+sin(x) is linear while u’’u + u’=x is non-linear.

Again, if p(x)=0 such an equation is said to be homogeneous. A solution of a homogeneous differential equation is called the homogeneous solution or complementary solution. When ( ) , the resulting solution is called the particular solution. The general solution of a differential equation is the sum of the homogeneous solution and the particular solution. A solution is not unique until the boundary conditions have been applied (or enforced) to evaluate the unknown constants in the solution. This requirement carries forward to when we replace the differential (strong) form with an integral (weak) form. The integral form solution is not unique until it satisfies the essential boundary conditions (EBC). Likewise, when finite elements

Parametric Finite Element Analysis J.E. Akin

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are employed to convert an integral form into a matrix form the matrix system is non-unique (singular) until it is modified to enforce the essential boundary conditions. Then the matrix solution can be solved and will yield the unique solution.

It is important to note that it can be proved that a finite element approximation of a differential equation will be exact at the nodes of the mesh if the piecewise spatial approximation (the element interpolation functions) contain at least the homogeneous solution. The solution at other points within the element usually will not be exact, but can be exact in some cases. If the element interpolation functions include at least the general solution then the results are exact everywhere in the element.

1.2.2 Boundary conditions

Boundary conditions are extremely important since they must be enforced to obtain a unique solution. There are two classes of boundary conditions; the essential (or Dirichlet) conditions and the non-essential boundary conditions. The essential boundary conditions assign known values to the solution, u(x, y, z), at specified regions on the boundary of the geometric shape, inside which the differential defines the solution. The non-essential boundary conditions (NBC) have two versions. The first is the Neumann condition which assigns known valves to the derivative of the solution normal (perpendicular) to the boundary like:

where b is the known boundary condition value, is the direction normal to the boundary, and denotes a known constitutive term component evaluated in that direction. In one-dimensional problems, keep in mind that at xmax the normal derivative of the solution is ⁄ ⁄ , but at xmin it changes sign to ⁄ ⁄ . The Neumann condition becomes a natural boundary condition when b=0, because it is the default (finite element) boundary condition on all boundary regions that do not have any other type of boundary condition specified.

The other non-essential boundary condition type is the mixed boundary condition (or Robins condition). In addition to the normal derivative, a mixed condition couples that to the unknown value of the solution at the boundary:

where c is a known boundary property (like a convection coefficient or elastic foundation stiffness).

Many engineering differential equations are even order (second or fourth). Let the even order be n=2m for , then the essential boundary conditions involve specifying the value of u and its (m-1) derivatives. The zero-th derivative is the function itself. For even order differential equations the non-essential boundary conditions involve the m-th to (2m-1) order derivatives. Two common applications to be studied later are listed here:

Application m n ODE EBC [0 to m-1] NBC [m to 2m-1]

Conduction 1 2 k u’’ = p u k u,n = q

Beam bending 2 4 EI u’’’’=p u and u,n EI u,nn=M and EI u,nnn=V

where ( ) ( ) ⁄ . In the case of heat conduction an example of a mixed boundary condition (surface convection) is: u,n + c u = b. An ODE of order n must have n boundary conditions applied. In theory, they can be any mixture of EBCs and NBCs. Applying NBCs only defines the solution to within an arbitrary constant. In that case, using a finite word length computer the resulting equations are singular and it is necessary to


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specify an arbitrary value of u on the interior of the region. Then the true solution is the computed one, relative to that given point, plus an arbitrary constant.

If a PDE is not even order the boundary conditions can be established by formulationg the adjoint of the PDE. For example, let a homogeneous PDE operator be represented as (u)=0, and form its inner product with another function, say v, to get

⟨ ( ) ⟩ ∫ ( )

.

Integrating by parts (sometimes repeatedly) gives an alternate weak form

⟨ ( ) ⟩ ⟨ ⟩ ∫ ( ) ( ) ( ) ( )

.

Here, and are differential operators that appear naturally from the integration(s) by parts process. The operator is called the adjoint of and is called self-adjoint if . Then, also. The ( ) terms are the essential boundary conditions on and the ( ) terms are the non-essential boundary conditions on . Self-adjoint operators always lead to a symmetric algebraic system of equations in FEA.

For an even order PDE, the highest derivative of u occurring in the variational or Galerkin integral form will also be of order m. If the highest derivative of u in any integral, over some region, is of order m then that integral can alternately be evaluated as the sum of integrals over the same region if, and only if, u and its (m-1) normal derivatives are continuous across the interface of the adjacent sub-regions.

In the integral form, the NBCs are carried into the integrals and the conversion to a matrix system places them in the column matrix with known values. If mixed NBC are present, then they also contribute known values to the square matrix of the algebraic system.

1.2.3 Related BCs

Due to laws of mechanics, physics or thermodynamics the combination of EBCs and NBCs is not completely arbitrary. As an example, consider the fourth-order beam equibrium ODE:

( ) ( )

( ) ( )

( ) ( )

( ) ( )

( ) .

Beam equilibrium can be stated as a work-energy principle. Mechanical work at a point can be written as the product of the transverse deflection and the transverse shear force and/or as the small slope, , multiplied by the moment. Therefore, at a BC point either ( ) ( ) can be specified as the EBC or NBC, but not both. Likewise, either ( ) ( ) can be given as the EBC or the NBC, but not both. When one of the two is specified as a boundary condition the other condition is computed from the corresponding integral (weak) form. When a NBC value is recovered by post-processing the integral form (after all the dependent variables have been computed) such a value is referred to as a reaction. In most engineering applications they have important physical meanings and can serve to check some input data. The same concept often applies to second order elliptic ODEs where if the EBC, ( ), is given then the NBC ( ) is computed as a reaction from post-processing the integral form.


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1.2.4 Classes of PDEs

The above general comments on EBC and NBC also apply to partial differential equations (PDEs). A typical second order two-dimensional partial differential equation, for any two independent variables, and with ( ) ( ) ⁄ and etc. is:

( ) ( ) ( ) ( ) ( ) ( ) ( ) .

The solutions, ( ), fall into three classes:

1. for elliptic PDEs. The solutions are as smooth as the coefficients interior to the solution region. Singular points may occur on the boundary. This book will mainly address even order elliptic PDEs.

2. for parabolic PDEs. Only parabolic problems advancing in time are covered in this book. Then, the EBC and NBC occur at the beginning of the time domain and are referred to as initial conditions (IC).

3. for hyperbolic PDEs. The solution and/or its derivatives can have discontinuities. Hyperbolic PDEs will not be considered in this book. A hyperbolic PDE typically addresses a wave propagation problem.

It is a property of all elliptic PDEs that the boundary of the solution domain often has points where the solution is singular. There the solution has an infinite derivative in the radial direction from the point ( ⁄ ), and tends to change rapidly in the transverse direction. Such points occur where there is a discontinuity in assigned EBC values, where an EBC jumps to an adjacent NBC and where the solution domain contains a sharp reentrant corner. The ‘strength’ of the singularity depends on the value of the radial exponent, ∑

( ). The first and most important term simply depends on the angle, interior to

the domain at the tip of the sharp corner, ⁄ . The strongest case is a crack so ⁄ . The most common case is a right angle corner ⁄ so ⁄ .

For example, a general two-dimensional PDE boundary value problem (BVP) in the region is:

(

)

(

) ( ) ( ) .

The essential boundary condition, on boundary region, , of the solution region, , is:

A non-essential boundary condition, on boundary region is:

(

) (

)

or

(

)

where the unit normal vector on the boundary is . This is often given as a natural boundary

condition, on , of:

(

)

or a mixed boundary condition, on , of:

(

)


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where the union of all the boundary regions forms the total boundary of the solution domain

.

1.2.5 Symmetry and anti-symmetry

There are many PDE’s where the region has one or more planes that describe a mirror image of the geometry and the material properties. If the solution also has a mirror image with respect to that plane then the analysis can be reduced to a symmetry analysis employing only the region of geometry symmetry. In that case, the edge of the material omitted must be subjected to a natural boundary condition of zero normal solution gradient along that line. In other cases, the geometry and material may be symmetric while the changes in the solution, on either side of the mirror plane, are the negative of each other. That leads to an anti-symmetric analysis. Symmetric models are very common. The use of each symmetry or anti-symmetry plane reduces the storage requirements by a factor of two and the matrix solution operations by as much as a factor of eight. These concepts are sketch in Figure 1.1.

Figure 1.1 Symmetry and anti-symmetry reduce storage and operation counts

1.3 Equivalent Integral Forms

1.3.1 Variational calculus

For elliptical differential equations there are two types of integral forms that have the same boundary conditions and whose solutions are also the exact solution of the PDE. Those two forms are Euler’s Theorem of Variational Calculus, and the Galerkin Weighted Residual Method, with integration by parts. Those two forms yield identical definitions for the finite element matrices. However, they involve distinctly different procedures. Variational calculus considers integrals of functionals (functions of functions).


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Euler (1750) presented integral forms that yield their equivalent PDE when they are minimized. In other words, Euler’s variational procedure starts with an integral and then establishes its corresponding PDE. Not all differential equations have a variational form, and nonlinear PDEs rarely have a variational form. The good news is that Euler gave the variational integral forms, and corresponding PDEs, for all the common linear even order elliptic PDEs of importance in engineering and physics.

There are several techniques known as Weighted Residual Methods (MWRs) which start with a PDE and create an equivalent integral form, and are valid for non-linear PDEs. The Galerkin process is a very popular weighted residual technique for several reasons. One reason is that by applying integration by parts (and Green’s theorem in two- or three-dimensions) the finite element matrices are exactly the same form as those as the variational integral form. Another reason is that it has a strong mathematical foundation. By starting with the concept of making the solution error orthogonal to the solution it can be shown that the Galerkin method gives the ’best’ approximation, as measured in a specific error norm.

For a 2m even order PDE the highest derivative in either of the variational and Galerkin integral forms is order m. Since the PDE requires a solution with derivatives of order 2m it is called the strong form. That contrasts with the equivalent integral form which requires the solution to be a function with derivatives of order m, which is called a weak solution. For the second order BVP given above, the equivalent variational form is:

∬

[ (

)

(

)

]

∬

∬

∫

∫ [

]

and the very similar Galerkin MWR integral is

∬ [ (

)

(

)

]

∬

∬

∫

∫

and both forms must satisfy the essential boundary condition on .

The process defined by Euler to determine the PDE and its boundary conditions begins with a functional, f, containing the independent variables, the solution and the solution’s partial derivatives with respect to the independent variables:

∬ (

) ∫ (

) .

To obtain a stationary value (here a minimum) the corresponding PDE in the region from Euler’s Theorem is

(

)

(

)

and the nonessential boundary condition, where u is not specified is:

.

The cases where c or b or both are zero would define different boundary regions and different NBCs.

In the above variational form:

,

,

.


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Thus, the corresponding PDE is

(

)

(

) .

1.3.2 Method of weighted residuals

See posted notes.

1.4 Calculus review

1.4.1 Change of variables

To be able to treat curvilinear regions with the finite element method it is necessary to define the physical space in terms of non-dimensional parametric coordinates. Therefore, the original governing integrals must employ a change of variable of integration and carry out the integrals in parametric space. Let x be the original independent spatial variable and replace it with ( ) where r is the choice for a new independent variable. Such a change is called a transformation or mapping of x, and ( ) ( ) is called a composite function. Then, it can be shown that

∫ ( )

∫ ( )

∫ ( )

( )

where ( ) and ( ) . The term ( ) ⁄ is called the determinant of the Jacobian of the transformation. In FEA, x has physical units and r is a non-dimensional parametric coordinate. Of course, the change of variables does not change the units of the original integral. Let be the original physical domain and let be a parametric space and define their measure (or generalized volume) as

| | ∫

( ) | | ∫

( ).

Here, x is typically a physical coordinate and ( ) would be a physical length while in the parametric space is a number giving the upper limit of the size of the space while is the lower limit of the parametric space. Here, usually while for unit parametric coordinates (and baracentric parametric coordinates) and for natural parametric coordinates. Thus, the parametric measure ( ) is just a non-dimensional number which is either 1 or 2 for one-dimensional problems.

In general, the Jacobian depends on the parametric coordinates, but it is often a constant. In the common special case of a constant Jacobian in one-dimension , the above integral (with ( ) ) gives

( )

( )

which clearly has the units of length.

1.4.2 Jacobian matrix

The differentiation of composite functions is required in every parametric finite element study. Let the dependent variable be u(x, y, z) with the transformations x=f(r, s, t), y=g(r, s, t) and z=h(r, s, t). From the chain rule of calculus the relationship between the parametric derivatives and the physical derivatives are:


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or in a matrix form this defines the Jacobian matrix, J(r, s, t),

{

}

[

]

{

}

( )

{

}

.

The revelance of different partitions of the full three-dimensional Jacobian matrix is denoted in Figure 1.2.

Figure 1.2 Partitions of the transformation Jacobian matrix

In the shorthand matrix notation of this book, the parametric partial derivatives are defined in terms of the physical partial derivatives symbolically as:

( )

( ) ( )

since the derivatives can be applied to any function, and not just u(x, y, z). The determinant of the Jacobian of the transformation, J=|[J]|, relates the physical differential volume and the differential parametric volume:

(

)

.

In practical FEA studies the Jacobian matrix and its (generalized) determinant are evaluated numerically. That is also true for the inverse of the Jacobian matrix.

The inverse of the Jacobian matrix is needed when the physical derivatives must be obtained from known parametric derivatives:

( ) ( ),


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{

}

[

]

{

}

.

In the one-dimensional case

⁄

.

For future analytic examples, the inverse Jacobian matrix in two-dimensions is

| |[

] (

).

Now, a typical change of variable of integration in two dimensions becomes

∬ ( )

∬ ( ) ( ) ( )

.

In FEA each element has a different Jacobian matrix. In other words, x(r) depends on the control point values on a particular element. Therefore, in FEA the Jacobian matrix is always element dependent and it is

denoted here as . Likewise, its determinant and inverse are denoted in a similar fashion as and .

1.4.3 Differential geometry

In some cases it is necessary to integrate along the edge of a curvilinear element or the face of a curvilinear solid. Then the concept of utilizing the determinant of the Jacobian to covert a parametric differential length to a physical differential length is generalized somewhat. Consider a planar curve where two physical coordinates are defined by a single parametric one with x=x(r) and y=y(r). A physical length increment

relates to the parametric increment, dr, as

(

)

(

)

(

)

or

√(

) (

) ( ) .

The tangent to a planar curve and the conversion ofr parametric differential length to a physical differential length aloing the planar curve are shown in Figure 1.3.

In FEA the NBC often require the identification of the vector normal to a surface and the integral over a non-flat surface. Parametric surfaces define the three components of the position vector in terms of the two parametric coordinates: x=x(r, s), y=y(r, s), z=z(r, s) and

( ) ( ) ( ) ( ) .

At any point on the surface there are two non-orthogonal vectors tangent to the surface. The tangent along r (at s constant) and along s are:

,


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Figure 1.3 Planar curve metric, J, and tangent vector, T

respectively, and the unit normal vector at the point is

| |

and the element of physical surface area is related to the parametric surface area by

( ) √(

) (

) (

)

so

( ) √( )( ) ( )

| |

noting that the tangent vectors are not unit vectors. That is, the tangent vectors have a magnitude and the units of length so the generalized Jacobian determinant, J(r, s), has the units of length squared and must be positive. If S is a surface on the boundary of a volumetric region , then any surface integral becomes

∬ ( )

∬

( ) ( ) ( ) ( )

or if the unit normal vector is included

∬ ( )

∬

( ) ( ) ( ) .

1.4.4 Integration by parts

The use of the popular Galerkin’s method generally requires the use of integration by parts, and Green’s Theorem in two- and three-dimensional space in order to reduce the continuity requirements of the weak solution and to render even order equations to a symmetric integral form. In one-dimension the relations are

∫ ( ) ( )

( ) ( )| ∫ ( ) ( )

∫ ( ) ( )

( ) ( )|

∫ ( ) ( )

.


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For a region bounded by the curves Green’s Theorem gives the following relations:

∭( )

∬

∬

∭( )

∬ (

)

∬ (

)

∮ ( )

.

Most of the time, this integrand must be equated to an item of interest. For example, equating it to the Laplacian:

(

)

gives ⁄ and ⁄ so

∮( )

∮ (

)

∮ (

)

∮( )

∮

therefore,

∬

∮

.

1.5 Axisymmetric integrals

Many finite element applications involve volumes and surfaces of revolution. Those problems are formulated in the radial and axial dimensions (that is the r-z plane), as a curve or closed area to be revolved about the z-axis. They are treated by the Theorem of Pappus. For an angle of revolution of the volume and surface area (excluding end faces when ) are

∫

∫

∫

∫

where R is the radial coordinate and is the radius to the shape centroid in the r-z plane. Of course, in finite element analysis the radial coordinate is interpolated from the radial coordinates of the nodes of the element.

1.6 Exact parametric integrals

When the Jacobian of a transformation is constant, it is possible to develop closed form exact integrals of parametric polynomials. Several examples of this will be presented in the next chapter for lines, triangles and rectangles. They are useful for simplifying lectures, but for practical finite element analysis, and its automation, it is necessary to employ numerical integration.

1.7 Numerical integration

Since the finite element method is based on integral relations it is logical to expect that one should strive to carry out the integrations as efficiently as possible. In some cases we will employ exact integration. In others


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we may find that the integrals can become too complicated to integrate exactly. In such cases the use of numerical integration will prove useful or essential. The important topics of local coordinate integration and Gaussian quadratures will be introduced here. They will prove useful when dealing with higher order interpolation functions in complicated element integrals.

Numerical integration is simply a procedure that approximates an integral by a summation. However, it can yield the exact intergral and often does when the integrand is a polynomial. The famous mathematician Gauss posed this question: What is the minimum number of points, , required to exactly integrate a

polynomial, and what are the corresponding abscissae and weights? If we require the summation to be exact when the integrand ( ) is any one of the power functions , we obtain a set of 2n conditions that allow us to determine the abscissae, , and their corresponding weights, . The

Gaussian quadrature points are symmetrically placed with respect to the center of the interval, and will exactly integrate a polynomial of order ( ). The center point location is included in the abscissae list when

is odd, but the end points are never utilized in a Gauss rule. The Gauss rule data are usually tabulated for a non-dimensional unit coordinate range of , or for a natural coordinate range of .

Tables are included in the text Matlab scripts for low-order Gauss rule data in natural coordinates, and in the alternate unit coordinate data. A two-point Gauss rule can often exceed the accuracy of a 20-point trapezoidal rule. When computing the norms of an exact solution, to be compared to the finite element solution, we often use the trapezoidal rule. That is because the exact solution is usually not a polynomial and the Gauss rule may not be accurate.

The quadrature rules for triangles do not have a simple equation relating the number of quadrature points to the polynomial degree. Instead, it is necessary to rely on published data which are given in Table 1.1. The location of the quadrature points in triangles fall along the lines that bisect the corner angles of the element, as shown in Figure 1.4.

Figure 1.4 Symmetric quadrature locations for unit triangle

Table 1.1 Selecting the quadrature rule for unit triangles

Polynomial degree 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17

Required points, 1, 3, 4, 6, 7, 12, 13, 16, 19, 25, 27, 33, 37, 42, 48, 52, 61


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Table 1.1 Abscissas and weights for Gaussian quadrature on -1 to 1

xi wi

0.00000 00000 00000 00000 0000 nq 1 2.00000 00000 00000 00000 000

0.57735 02691 89625 76450 9149 nq 2 1.00000 00000 00000 00000 000

0.77459 66692 41483 37703 5835 nq 3 0.55555 55555 55555 55555 556

0.00000 00000 00000 00000 0000 0.88888 88888 88888 88888 889

0.86113 63115 94052 57522 3946 nq 4 0.34785 48451 37453 85737 306

0.33998 10435 84856 26480 2666 0.65214 51548 62546 14262 694

0.90617 98459 38663 99279 7627 nq 5 0.23692 68850 56189 08751 426

0.53846 93101 05683 09103 6314 0.47862 86704 99366 46804 129

0.00000 00000 00000 00000 0000 0.56888 88888 88888 88888 889

0.93246 95142 03152 02781 2302 nq 6 0.17132 44923 79170 34504 030

0.66120 93864 66264 51366 1400 0.36076 15730 48138 60756 983

0.23861 91860 83196 90863 0502 0.46791 39345 72691 04738 987

0.94910 79123 42758 52452 6190 nq 7 0.12948 49661 68869 69327 061

0.74153 11855 99394 43986 3865 0.27970 53914 89276 66790 147

0.40584 51513 77397 16690 6607 0.38183 00505 05118 94495 037

0.00000 00000 00000 00000 0000 0.41795 91836 73469 38775 510

See tabulated values in function qp_rule_Gauss.m

Table 1.2 Unit abscissas and weights for Gaussian quadrature on 0 to 1

xi wi

0.50000 00000 00000 00000 000 nq 1 1.00000 00000 00000 00000 000

0.21132 48654 05187 11774 543 nq 2 0.50000 00000 00000 00000 000

0.78867 51345 94812 88225 457 0.50000 00000 00000 00000 000

0.11270 16653 79258 31148 208 nq 3 0.27777 77777 77777 77777 778

0.50000 00000 00000 00000 000 0.44444 44444 44444 44444 444

0.88729 83346 20741 68851 792 0.27777 77777 77777 77777 778

0.06943 18442 02973 71238 803 nq 4 0.17392 74225 68726 92868 653

0.33000 94782 07571 86759 867 0.32607 25774 31273 07131 347

0.66999 05217 92428 13240 133 0.32607 25774 31273 07131 347

0.93056 81557 97026 28761 197 0.17392 74225 68726 92868 653

0.04691 00770 30668 00360 119 nq 5 0.11846 34425 28094 54375 713

0.02307 65344 94715 84544 818 0.23931 43352 49683 23402 065

0.50000 00000 00000 00000 000 0.28444 44444 44444 44444 444

0.76923 46550 52841 54551 816 0.23931 43352 49683 23402 065

0.95308 99229 69331 99639 881 0.11846 34425 28094 54375 713

See tabulated values in function qp_rule_unit_Gauss.m


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Table 1.3 Abscissas and weights for Lobatto integration on -1 to 1

xi wi

0.00000 00000 00000 nq 1 2.00000 00000 00000

1.00000 00000 00000 nq 2 1.00000 00000 00000

1.00000 00000 00000 nq 3 0.33333 33333 33333

0.00000 00000 00000 1.33333 33333 33333

1.00000 00000 00000 nq 4 0.16666 66666 66667

0.44721 35954 99958 0.83333 33333 33333

1.00000 00000 00000 nq 5 0.10000 00000 00000

0.65465 36707 07977 0.54444 44444 44444

0.00000 00000 00000 0.71111 11111 11111

1.00000 00000 00000 nq 6 0.06666 66666 66667

0.76505 53239 29465 0.37847 49562 97847

0.28523 15164 80645 0.55485 83770 35486

Table 1.4 Symmetric quadrature for the unit triangle:

n p† i ri si Wi

1 1 1 1/3 1/3 1/2

3 2 1 1/6 1/6 1/6

2 2/3 1/6 1/6

3 1/6 2/3 1/6

4 3 1 1/3 1/3 −9/32

2 3/5 1/5 25/96

3 1/5 3/5 25/96

4 1/5 1/5 25/96

7 4 1 0 0 1/40

2 1/2 0 1/15

3 1 0 1/40

4 1/2 1/2 1/15

5 0 1 1/40

6 0 1/2 1/15

7 1/3 1/3 9/40

p = Degree of polynomial for exact integration.

See tabulated values in function qp_rule_unit_tri.m


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Table 1.5 Gaussian quadrature on a quadrilateral

n i ri si Wi

1 1 0 0 4

4 1 −1/3 −1/3 1 2 +1/3 −1/3 1 3 −1/3 +1/3 1 4 +1/3 +1/3 1

9 1 −3/5 −3/5 25/81 2 0 −3/5 40/81 3 +3/5 −3/5 25/81 4 −3/5 0 40/81 5 0 0 64/81 6 +3/5 0 40/81 7 −3/5 +3/5 25/81 8 0 +3/5 40/81 9 +3/5 +3/5 25/81

See computed values in function qp_rule_nat_quad.m

The quadrature rules for tetrahedra do not have a simple equation relating the number of quadrature points to the polynomial degree. Instead, it is necessary to rely on published data which are given in Table 1.6. The location of the quadrature points in tetrahedral fall along the lines that run from the corner vertex to perpendicular to the opposite face.

Table 1.6 Selecting the quadrature rule for a unit tetrahedra

Polynomial degree 0, 1, 2, 3, 4, 5, 6, 7, 8

Required points, 1, 4, 5, 10, 11, 15, 24, 31, 45


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Table 1.6 Quadrature for unit tetrahedra

Number

of

points N

Degree

of

precision

Unit coordinates

ri si ti

Weights

Wi

1 1 1/4 1/4 1/4 1/6

4 2 a b b 1/24

b a b 1/24

b b a 1/24

b b b 1/24

a ( 5 35 ) / 20 0. 5854101966249685

b ( 5 5 ) / 20 0. 1381966011250105

5 3 1/4 1/4 1/4 − 4/30

1/2 1/6 1/6 9/120

1/6 1/2 1/6 9/120

1/6 1/6 1/2 9/120

1/6 1/6 1/6 9/120

11 4 1/4 1/4 1/4 − 74/5625

11/14 1/14 1/14 343/45000

1/14 11/14 1/14 343/45000

1/14 1/14 11/14 343/45000

1/14 1/14 1/14 343/45000

a a b 56/2250

a b a 56/2250

a b b 56/2250

b a a 56/2250

b a b 56/2250

b b a 56/2250

a ( 1 (5/14) ) / 4 0. 3994035761667992

b ( 1 (5/14) ) / 4 0. 1005964238332008

1.8 Polynomial interpolation

1.8.1 Lagrange and Serendipity interpolation

Interpolation requires estimating the values of a function u(r) based on locations for at which the values are known. In this book, only equally spaced intervals will be employed in the parametric space. The expressions for the finite element interpolations are well known and will generally be given without proof. For completeness, the one-dimensional quadratic interpolation function, in unit coordinates will be derived. A quadratic polynomial in one-dimension has three constants. The three parametric locations are ,


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⁄ and , so in the above notation and and the measure of the parametric range is ( ) . Let the polynomial data fit be

( ) {

} ( ) ( )

To define the three consants evaluate this assumption at the three parametric locations to yield a matrix identity:

{

( ) ( ) ( )

} {

} [

] {

} .

The square matrix, , has only known numbers from the geometry of the parametric interval. Here, it is non-singular and its inverse is computable, and lets the initial constants be expressed interms of the equally spaced function values:

.

Eliminating the non-physical constants in favor of the interpolation data yields the interpolation equations, ( )also known as the shape functions in FEA:

( ) ( ) ( )

( ) [

] {

}

( ) ( ) ( ) ( ) {

} ( ) ( ) ( ) {

}.

Having made this selection for the parametric spatial form the parametric local derivative is also known as:

( ) ( ) ( ) ( ) {

}

( )

( )

( ) {

}.

Since this interpolation approach exactly matches the data, , at the control points, , this is a Lagrangian interpolation. In this book, linear through cubic polynomials will be applied to second order PDEs and cubic through quartic polynomials will be utilized in fourth order PDEs. In general, a Lagrange interpolation function evaluated at its control point will have a value of unity and it will have a value of zero at any other control point:

( ) {

.

1.8.2 Quadrilaterial and hexahedra interpolations

For quadrilaterials and hexahedra (bricks) their Lagrange interpolations can be obtained by the same process outlined above by solving a small sysyem of linear equations. When the parametric nodes are equally spaced then the required matrix inversion usually succeeds. The quadrilaterial and hexahedra interpolations can also be obtained directly by taking the tensor product of the one-dimensional interpolations in each of the local coordinate directions. Usually, that produces control points (nodes) interior to the parametric domain. Similar interpolations that utilize only control points on the edges of the parametric space are known as Serendipity interpolations.


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Here, the interpolation functions for the Lagrange bi-linear four-node quadrilaterial (Q4) will be developed, in unit coordinates. From Figure 1.5 the first two nodes on the quadrilateral have the interpolations at s = 0 multiplied times the two r-interpolations, and so on:

( ) ( ) ( ) ( )( )

( ) ( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( ) ( )

and ∑ ( ) , as expected. This form is useful for lectures, but most quadrilaterials and hexahedra are

formulated in natural coordinates, ( ) . In natural coordinates the above interpolation functions for node k can be expressed in a simple form as

( )

( )( )

where ( ) are the parametric coordinates of the node. For the eight node hexahedra, H8, element this exspands to

( )

( )( )( )

Several of the most common Lagrangian and Serendipity interpolations are included in function el_shape_n_local_deriv.m.

Figure 1.5 Bilinear quadrilateral formed by two linear products

1.8.3 Triangular and tetrahedra interpolations

For triangles and tetrahedra their Lagrange interpolations can be obtained by the same process outlined above by solving a small sysyem of linear equations. When the parametric nodes are equally spaced then the required matrix inversion usually succeeds. These elements are usually expressed in unit coordinates or baracentric (area or volume) coordinates. For the complete linear three-node triangle (T3) in unit coordinates the interpolation functions at (0, 0), (1, 0) and (0, 1) are:

( )

( )

( )

∑ ( ) .


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The complete quadratic six-node triangle (T6) interpolations, numbered CCW corners then midsides, are:

( )

( )

( )

( )

( )

( ) .

Since the symbol T# is being used to indicate specific triangular element interpolation, the tetrahedral elements will be denoted as pyramids with a P# symbol. The complete linear four-node tetrahedral, P4, shown in Figure 1.6, in unit coordinates has the following interpolation functions:

( )

( )

( )

( )

Figure 1.6 Linear tetrahedron simplex element

This element has straight edges and flat faces and clearly has a constant Jacobian matrix. The parametric volume, from geometry, is half the base area times the height, | | ( ⁄ )( ⁄ ) ⁄ , so the previous procedure will yield the determinant of the Jacobian as | | | |⁄ ( ⁄ ) ⁄ . A quadratic tetrahedral, P10, will generally have a variable Jacobian, but will degenerate to the same constant values if the geometry data gives it straight edges and thus flat faces.

The solid elements on their faces degenerate to the corresponding parametric area element and the area elements degenerate on their edges to the corresponding parametric line elements. That is one way that the

boundary interpolations, can be reduced to its sub-set boundary interpolation, . This can be seen in the T4 element by restricting it to t = 0, which is the r-s plane.:

( )


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( )

( )

( )

Note that all nodes not on that face , in this case node 4, contribute nothing to the interpolation and that the non-zero contributions become the three node triangle given previously. Likewise, if the interpolation is restricted to the edge where s = 0, t = 0, that is the r-axis, the T4 interpolation reduces to

( )

( )

( )

( )

and the contribution of any node not on that edge becomes zero.

1.8.4 Hermite interpolation

Polynomials that exactly interpolate the data, , and the physical derivatives at the control points,

⁄ , define Hermite interpolation. In that case, there are two degrees of freedom at each control point (node). The above three-node parametric space would have six constants so the polynomial must be of fifth degree. Since the physical slope is included in the control data it is necessary to also define x(r). Since x is a scalar, use the same quadratic interpolation process and set ( ) ( ) . Usually, the physical nodes are also equally spaced. In that case, ⁄ and , where is the length of the region in physical space. Then, the quadratic terms cancel and the geometry mapping is linear ( )

( ) ( ) so ( ) ⁄⁄⁄ .

Repeating the above process and inverting the 6x6 matrix gives the three-node quintic Hermite interpolation:

( ) ( ) ( )

( ) ( )

( ) ( )

and its slope:

( ) ( ) ⁄ ( ) ( )

( ) ( ) ( )

( ) .

Several of the most common Hermite interpolations are included in function el_shape_n_global_deriv_C1.m.

1.9 Coupled matrix partitions

In courses on matrix algebra the linear systems usually involve non-singular square matrices. That is because any constraints on the unknowns were satisfied in advance. In the finite element applications to be covered here the constraints are satisfied last, which results in an initially singular square matrix because the reactions due to the cinstraints appear as unknowns on the column vector side of the equations. In other words, because the essential boundary conditions will be applied last and there is not a unique solution until that is done. A matrix partition must be used to allow for the solution constraints and to yield a smaller and non-


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u uu u

singular system of equations to be solved. The algebraic system can be written in a general matrix form that more clearly defines what must be done to reduce the system to a solvable form by utilizing essential boundary condition values. Note that the system degrees of freedom, D, and the full equations could always be re- arranged in the following partitioned matrix form

[

] {

} {

}

where represents the unknown nodal parameters, and represents the known essential boundary

values of the other parameters. The sub-matrices and are square, whereas and are

rectangular, in general. In a finite element formulation all of the coefficients in the S and C matrices are

known. The term represents that there are usually unknown generalized reactions associated with

essential boundary conditions. This means that in general after the essential boundary conditions ( ) are

prescribed the remaining unknowns are and . Then the net number of unknowns corresponds to the

number of equations, but they must be re-arranged before all the remaining unknowns can be computed.

Moving the unknown reactions, , to the left side would destroy any usefully symmetry in the square

matrix, so instead they are obtained as a post solution calculation.

Here, for simplicity, it has been assumed that the equations have been numbered in a manner that places the

prescribed parameters (essential boundary conditions) at the end of the system equations. The above matrix

equations can be re-written in expanded form as:

so that the unknown nodal parameters are obtained by inverting the non-singular square matrix

in the top partitioned rows. That is,

( ).

Most books on numerical analysis assume that you have reduced the system to the above non- singular form

where the essential conditions, , have already been moved to the right hand side. Many authors use

examples with null conditions so the solution is the simplest form, . If desired, the

values of the necessary reactions, , can now be determined from

In nonlinear and time dependent applications the reactions can be found from similar calculations. In most applications the reaction data have physical meanings that are important in their own right, or useful in validating the solution. However, this part of the calculations is optional. If one formulates a finite element model that satisfies the essential boundary conditions in advance, then the second row of the partitioned system S matrix is usually not generated and one can not recover the reaction data. The above matrix partitioning can be accomplished easily with languages that allow vector subscripts. Vector subscripts are efficient programming tools in FEA. They are standard in the Fortran language and in the Matlab environment. Figure 1.7 illustrates how vector subscripts can extract the above partitions in a coupled matrix system and use them to solve for the unkowns and the reactions from the EBC. However, the sub-matrix partitions take up temporary memory locations and since S is usually huge that can be a concern.

The actual partitioning of the full matrix is not actually necessary and involves some computational cost. Assigning known value to a single degree of freedom, say , can be done with numerical manipulations. Since the number of essential boundary conditions are often a very small percentage of the unknowns the


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enforcement of the EBC can be completed by a matrix operation that modifies both the square matrix and the column vector for each assigned value in D. The operation first subtracts the assigned value, , times the m-th column of S, and then zeros out that column along with the m-th row of both S and C. Finally, it inserts a scalar identity on the diagonal and in the column vector, , so the size of the equations remains the same but the square matrix is rendered non-singular. For the complete details see the function enforce_essential_BC and/or the related penalty method in enforce_MPC_equations. If the reaction data, , need to be recovered in post-processing, then the lower partition of S and C must be saved for later use.

Figure 1.7 Matrix partitions using vector subscripts (Free, Fixed)

In practical FEA the size of the equations is so large that a matrix inversion is not practical and a factorization of the equations is used. That is especially efficient when the square matrix is symmetric and positive definite. When the matrix size gets extremely large an iterative solver, like a pre-conditioned conguant gradient


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method, is utilized. The process for factoring a square matric into an upper and lower triangle product is sketched in Figure 1.8. After the factorization, a forward substitution loop and a backward substitution loop are run. They each involve only one equation and one unknown when solving for the intermidate dummy unknown and the actual unknown in each row of the matrix system.


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Figure 1.8 Utilizing matrix factorization instead of matrix inversion

1.10 Exercises

1 mathematical review - rice university

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