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Finite Element Approximation

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___ Introduction

Our goal in this chapter is the development of piecewise_polynomial approximations U

of a two_ or three_dimensional function u_ For this purpose_ it su_ces to regard u as

being known and to determine U as its interpolant on a domain __ Concentrating on

two dimensions for the moment_ let us partition _ into a collection of _nite elements and

write U in the customary form

U_x_ y_

Xj__

cj_j_x_ y__ ______

As we discussed_ it is convenient to associate each basis function _j with a mesh entity_

e_g__ a vertex_ edge_ or element in two dimensions and a vertex_ edge_ face_ or element

in three dimensions_ We will discuss these entities and their hierarchical relationship

further in Chapter __ For now_ if _j is associated with the entity indexed by j_ then_ as

described in Chapters _ and _ _nite element bases are constructed so that _j is nonzero

only on elements containing entity j_ The support of two_dimensional basis functions

associated with a vertex_ an edge_ and an element interior is shown in Figure _____

As in one dimension_ _nite element bases are constructed implicitly in an element_

by_element manner in terms of _shape functions_ _cf_ Section ___ Once again_ a shape

function on an element e is the restriction of a basis function _j_x_ y_ to element e_

We proceed by constructing shape functions on triangular elements _Section __ ___

quadrilaterals _Sections ___ ___ tetrahedra _Section ______ and hexahedra _Section

_____

Finite Element Approximation

_____________

________

_______

Figure _____ Support of basis functions associated with a vertex_ edge_ and element

interior _left to right__

___ Lagrange Shape Functions on Triangles

Perhaps the simplest two_dimensional Lagrangian _nite element basis is a piecewise_linear

polynomial on a grid of triangular elements_ It is the two_dimensional analog of the hat

functions introduced in Section ____ Consider an arbitrary triangle e with its vertices

indexed as __ _ and _ and vertex j having coordinates _xj_ yj__ j __ _ _ _Figure _____

The linear shape function Nj_x_ y_ associated with vertex j satis_es

Nj_xk_ yk_ _j_k_ j_ k __ _ __ _____

_Again_ we omit the subscript e from Nj_e whenever it is clear that we are discussing a

single element__ Let Nj have the form

Nj_x_ y_ a _ bx _ cy_ _x_ y_ _ _e_

where _e is the domain occupied by element e_ Imposing conditions _____ produces

_ xj yj

_ xk yk

_ xl yl

_ k _ l _ j_ j_ k_ l __ _ __

Solving this system by Crammer_s rule yields

Nj_x_ y_

Dk_l_x_ y_

Cj_k_l

_ k _ l _ j_ j_ k_ l __ _ __ ___a_

where

Dk_l det_

_ x y

_ xk yk

_ xl yl

_ ___b_

____ Lagrange Shape Functions on Triangles _

(x ,y )

1 1

2 2

3 3

Figure ____ Triangular element with vertices __ __ having coordinates _x__ y___ _x__ y___

and _x__ y___

1

Figure ___ Shape function N_ for Node _ of element e _left_ and basis function __ for

a cluster of four _nite elements at Node __

Cj_k_l det_

_ xj yj

_ xk yk

_ xl yl

_ ___c_

Basis functions are constructed by combining shape functions on neighboring elements

as described in Section __ A sample basis function for a four_element cluster is shown in

Figure ___ The implicit construction of the basis in terms of shape function eliminates

the need to know detailed geometric information such as the number of elements sharing

Finite Element Approximation

a node_ Placing the three nodes at element vertices guarantees a continuous basis_ While

interpolation at three non_colinear points is _necessary and_ su_cient to determine a

unique linear polynomial_ it will not determine a continuous approximation_ With vertex

placement_ the shape function _e_g__ Nj_ along any element edge is a linear function of

a variable along that edge_ This linear function is determined by the nodal values at

the two vertex nodes on that edge _e_g__ j and k__ As shown in Figure ___ the shape

function on a neighboring edge is determined by the same two nodal values_ thus_ the

basis _e_g__ _j_ is continuous_

The restriction of U_x_ y_ to element e has the form

U_x_ y_ c_N__x_ y_ _ c_N__x_ y_ _ c_N__x_ y__ _x_ y_ _ _e_ _____

Using ______ we have cj U_xj_ yj__ j __ _ __

The construction of higher_order Lagrangian shape functions proceeds in the same

manner_ In order to construct a p th_degree polynomial approximation on element e_ we

introduce Nj_x_ y__ j __ _ _ _ _ _ np_ shape functions at np nodes_ where

_p _ ___p _ _

____

is the number of monomial terms in a complete polynomial of degree p in two dimensions_

We may write a shape function in the form

Nj_x_ y_

Xi__

aiqi_x_ y_ aTq_x_ y_ ____a_

where

qT _x_ y_ ___ x_ y_ x__ xy_ y__ _ _ _ _ yp__ ____b_

Thus_ for example_ a second degree _p _ polynomial would have n_ _ coe_cients

and

qT _x_ y_ ___ x_ y_ x__ xy_ y___

Including all np monomial terms in the polynomial approximation ensures isotropy in the

sense that the degree of the trial function is conserved under coordinate translation and

rotation_

With six parameters_ we consider constructing a quadratic Lagrange polynomial by

placing nodes at the vertices and midsides of a triangular element_ The introduction of

nodes is unnecessary_ but it is a convenience_ Indexing of nodes and other entities will be

discussed in Chapter __ Here_ since we_re dealing with a single element_ we number the

____ Lagrange Shape Functions on Triangles _

4 5

8 7

9 10

Figure ____ Arrangement of nodes for quadratic _left_ and cubic _right_ Lagrange _nite

element approximations_

nodes from _ to _ as shown in Figure ____ The shape functions have the form _____

with n_ _

Nj a_ _ a_x _ a_y _ a_x_ _ a_xy _ a_y__

and the six coe_cients aj _ j __ _ _ _ _ _ __ are determined by requiring

Nj_xk_ yk_ _j_k_ j_ k __ _ _ _ _ _ __

The basis

_j _N_

e__Nj_e_x_ y_

is continuous by virtue of the placement of the nodes_ The shape function Nj_e is a

quadratic function of a local coordinate on each edge of the triangle_ This quadratic

function of a single variable is uniquely determined by the values of the shape functions

at the three nodes on the given edge_ Shape functions on shared edges of neighboring

triangles are determined by the same nodal values_ hence_ ensuring that the basis is

globally of class C__

The construction of cubic approximations would proceed in the same manner_ A

complete cubic in two dimensions has __ parameters_ These parameters can be deter_

mined by selecting __ nodes on each element_ Following the reasoning described above_

we should place four nodes on each edge since a cubic function of one variable is uniquely

determined by prescribing four quantities_ This accounts for nine of the ten nodes_ The

last node can be placed at the centroid as shown in Figure ____

The construction of Lagrangian approximations is straight forward but algebraically

complicated_ Complexity can be signi_cantly reduced by using one of the following two

coordinate transformations_

_ Finite Element Approximation

__________________________

_ _________

_________

3 (x ,y )

1 (x ,y )

1 1

3 3

1 (0,0) 2 (1,0)

3 (0,1)





N = 0

3 N = 1

2 (x ,y ) 2 2

N = 0

N 1 = 1

Figure ___ Mapping an arbitrary triangular element in the _x_ y__plane _left_ to a

canonical __ right triangle in the ___ ___plane _right__

__ Transformation to a canonical element_ The idea is to transform an arbitrary

element in the physical _x_ y__plane to one having a simpler geometry in a computational

___ ___plane_ For purposes of illustration_ consider an arbitrary triangle having vertex

nodes numbered __ _ and _ which is mapped by a linear transformation to a unit __

right triangle_ as shown in Figure ___

Consider N_

_ and N_

_ as de_ned by _____ _A superscript _ has been added to

emphasize that the shape functions are linear polynomials__ The equation of the line

connecting Nodes _ and _ of the triangular element shown on the left of Figure __ is

_ __ Likewise_ the equation of a line passing through Node and parallel to the

line passing through Nodes _ and _ is N_

_ __ Thus_ to map the line N_

_ _ onto the

line _ _ in the canonical plane_ we should set _ N_

_ _x_ y__ Similarly_ the line joining

Nodes _ and satis_es the equation N_

_ __ We would like this line to become the line

_ _ in the transformed plane_ so our mapping must be _ N_

_ _x_ y__ Therefore_ using

____

_ N_

_ _x_ y_

det _

_ x y

_ x_ y_

det _

_ x_ y_

_ _ N_

_ _x_ y_

det _

_ x y

_ x_ y_

det _

_ x_ y_

_ _____

As a check_ evaluate the determinants and verify that _x__ y__ _ ___ ___ _x__ y__ _ ___ ___

and _x__ y__ _ ___ ___

Polynomials may now be developed on the canonical triangle to simplify the algebraic

____ Lagrange Shape Functions on Triangles _

N = 0

N = 2/3 N = 1/3

10 1

Figure ____ Geometry of a triangular _nite element for a cubic polynomial Lagrange

approximation_

complexity and subsequently transformed back to the physical element_

__ Transformation using triangular coordinates_ A simple procedure for constructing

Lagrangian approximations involves the use of a redundant coordinate system_ The

construction may be described in general terms_ but an example su_ces to illustrate the

procedure_ Thus_ consider the construction of a cubic approximation on the triangular

element shown in Figure ____ The vertex nodes are numbered __ _ and __ edge nodes

are numbered to __ and the centroid is numbered as Node ___

Observe that

_ the line N_

_ _ passes through Nodes _ __ __ and __

_ the line N_

_ ___ passes through Nodes __ ___ and __ and

_ the line N_

_ __ passes through Nodes and __

Since N_

_ must vanish at Nodes _ __ and be a cubic polynomial_ it must have the form

_ _x_ y_ _N_

_ _N_

_ _ _____N_

_ _ ___

where the constant _ is determined by normalizing N_

_ _x__ y__ __ Since N_

_ _x__ y__ __

we _nd _ __ and

_ _x_ y_

_ _N_

_ _ _____N_

_ _ ____

The shape function for an edge node is constructed in a similar manner_ For example_

in order to obtain N_

_ we observe that

_ the line N_

_ _ passes through Nodes __ __ __ and __

_ Finite Element Approximation

_ the line N_

_ _ passes through Nodes _ __ __ and __ and

_ the line N_

_ ___ passes through Nodes __ ___ and __

Thus_ N_

_ must have the form

_ _x_ y_ _N_

_N_

_ _N_

_ _ _____

Normalizing N_

_ _x__ y__ _ gives

_ _x__ y__ _

_ _

Hence_ _ __ and

_ _x_ y_

_N_

_ _N_

_ _ _____

Finally_ the shape function N_

__ must vanish on the boundary of the triangle and is_

thus_ determined as

___x_ y_ _N_

_N_

_ _

The cubic shape functions N_

_ _ N_

__ and N_

__ are shown in Figure ____

The three linear shape functions N_

j _ j __ _ __ can be regarded as a redundant

coordinate system known as _triangular_ or _barycentric_ coordinates_ To be more

speci_c_ consider an arbitrary triangle with vertices numbered __ _ and _ as shown

in Figure ____ Let

_ N_

_ _ _ N_

_ _ _____

and de_ne the transformation from triangular to physical coordinates as

x_ x_ x_

y_ y_ y_

_ _ _

_ _____

Observe that _ __ __ __ has value _______ at vertex __ _______ at vertex and _______ at

vertex __

An alternate_ and more common_ de_nition of the triangular coordinate system in_

volves ratios of areas of subtriangles to the whole triangle_ Thus_ let P be an arbitrary

point in the interior of the triangle_ then the triangular coordinates of P are

AP__

A___

_ _

AP__

A___

_ _

AP__

A___

_ _____

where A___ is the area of the triangle_ AP__ is the area of the subtriangle having vertices

P_ _ __ etc_

____ Lagrange Shape Functions on Triangles _

0.2

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0.6

0.8

1 0

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0.6

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1 0

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Figure ____ Cubic Lagrange shape functions associated with a vertex _left__ an

edge_right__ and the centroid _bottom_ of a right __ triangular element_

The triangular coordinate system is redundant since two quantities su_ce to locate

a point in a plane_ This redundancy is expressed by the third of equations ______ which

states that

_ _ _ _ _ __

This relation also follows by adding equations ______

Although seemingly distinct_ triangular coordinates and the canonical coordinates are

closely related_ The triangular coordinate _ is equivalent to the canonical coordinate _

and _ is equivalent to __ as seen from _____ and ______

Problems

__ With reference to the nodal placement and numbering shown on the left of Figure

____ construct the shape functions for Nodes _ and of the quadratic Lagrange

polynomial_ Derive your answer using triangular coordinates_ Having done this_

also express your answer in terms of the canonical ___ __ coordinates_ Plot or sketch

__ Finite Element Approximation

3 (0,0,1)

2 (0,1,0)

1 (1,0,0) P( 1 2 3



1 

Figure ____ Triangular coordinate system_

the two shape functions on the canonical element_

_ A Lagrangian approximation of degree p on a triangle has three nodes at the vertices

and p _ _ nodes along each edge that are not at vertices_ As we_ve discussed_

the latter placement ensures continuity on a mesh of triangular elements_ If no

additional nodes are placed on edges_ how many nodes are interior to the element

if the approximation is to be complete_

___ Lagrange Shape Functions on Rectangles

The triangle in two dimensions and the tetrahedron in three dimensions are the poly_

hedral shapes having the minimum number of edges and faces_ They are optimal for

de_ning complete C_ Lagrangian polynomials_ Even so_ Lagrangian interpolants are

simple to construct on rectangles and hexahedra by taking products of one_dimensional

Lagrange polynomials_ Multi_dimensional polynomials formed in this manner are called

_tensor_product_ approximations_ we_ll proceed by constructing polynomial shape func_

tions on canonical _ square elements and mapping these elements to an arbitrary

quadrilateral elements_ We describe a simple bilinear mapping here and postpone more

complex mappings to Chapter __

We consider the canonical _ square f___ __j__ _ __ _ _ _g shown in Figure _____

For simplicity_ the vertices of the element have been indexed with a double subscript

as ___ ___ __ ___ ___ __ and __ __ At times it will be convenient to index the vertex

coordinats as __ ___ __ __ __ ___ and __ __ With nodes at each vertex_ we

construct a bilinear Lagrangian polynomial U___ __ whose restriction to the canonical

____ Lagrange Shape Functions on Rectangles __

1,1

y 1,2 1,2

2,1

2,2 3,2 2,2

1,1 3,1 2,1

1,3 3,3 2,3

Figure _____ Node indexing for canonical square elements with bilinear _left_ and bi_

quadratic _right_ polynomial shape functions_

element has the form

U___ __ c___N______ __ _ c___N______ __ _ c___N______ __ _ c___N______ ___ _____a_

As with Lagrangian polynomials on triangles_ the shape function Ni_j___ __ satis_es

Ni_j__k_ _l_ _i_k_j_l_ k_ l __ _ _____b_

Once again_ U__k_ _l_ ck_l_ however_ now Ni_j is the product of one_dimensional hat

functions

Ni_j___ __ _N

i___ _N

j___ _____c_

with

____

_ _ _

_ _____d_

____

_ _ _

_ __ _ _ _ __ _____e_

Similar formulas apply to _N

j____ j __ _ with _ replaced by _ and i replaced by j_

The shape function N___ is shown in Figure ____ By examination of either this _gure or

_____c_e__ we see that Ni_j___ __ is a bilinear function of the form

Ni_j___ __ a_ _ a__ _ a__ _ a____ __ _ __ _ _ __ _____

The shape function is linear along the two edges containing node _i_ j_ and it vanishes

along the two opposite edges_

A basis may be constructed by uniting shape functions on elements sharing a node_

The piecewise bilinear basis functions _i_j when Node _i_ j_ is at the intersection of four

_ Finite Element Approximation

−1

−0.5

0.5

1 −1

−0.5

0.5

0.2

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0.6

0.8

−1

−0.5

0.5

1 −1

−0.5

0.5

0.2

0.4

0.6

0.8

Figure ____ Bilinear shape function N___ on the ____ _______ __ canonical square element

_left_ and bilinear basis function at the intersection of four square elements _right__

square elements is shown in Figure ____ Since each shape function is a linear polynomial

along element edges_ the basis will be continuous on a grid of square _or rectangular_ ele_

ments_ The restriction to a square _or rectangular_ grid is critical and the approximation

would not be continuous on an arbitrary mesh of quadrilateral elements_

To construct biquadratic shape functions on the canonical square_ we introduce _

nodes_ ______ _____ ____ and ____ at the vertices_ ______ _____ _____ and _____ at mid_

sides_ and _____ at the center _Figure ______ The restriction of the interpolant U to this

element has the form

U___ __

Xi__

Xj__

ci_jNi_j___ __ _____a_

where the shape functions Ni_j _ i_ j __ _ __ are products of the one_dimensional quadratic

polynomial Lagrange shape functions

Ni_j___ __ _N

i___ _N

j____ i_ j __ _ __ _____b_

with _cf_ Section __

____ ____ _ ____ _____c_

____ ___ _ ____ _____d_

____ __ _ ____ __ _ _ _ __ _____e_

Shape functions for a vertex_ an edge_ and the centroid are shown in Figure _____

Using _____b_e__ we see that shape functions are biquadratic polynomials of the form

Ni_j___ __ a_ _ a__ _ a__ _ a___ _ a___ _ a___ _ a ___ _ a___ _ a______ _____

____ Lagrange Shape Functions on Rectangles __

−1

−0.5

0.5

1 −1

−0.5

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0.2

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−0.5

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1 −1

−0.5

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Figure _____ Biquadratic shape functions associated with a vertex _left__ an edge _right__

and the centroid _bottom__

Although _____ contains some cubic and quartic monomial terms_ interpolation accuracy

is determined by the highest_degree complete polynomial that can be represented exactly_

which_ in this case_ is a quadratic polynomial_

Higher_order shape functions are constructed in similar fashion_

_____ Bilinear Coordinate Transformations

Shape functions on the canonical square elements may be mapped to arbitrary quadri_

laterals by a variety of transformations _cf_ Chapter ___ The simplest of these is a

picewise_bilinear function that uses the same shape functions _____d_e_ as the _nite el_

ement solution _____a__ Thus_ consider a mapping of the canonical _ square S to

a quadrilateral Q having vertices at _xi_j_ yi_j__ i_ j __ _ in the physical _x_ y__plane

_Figure ____ using a bilinear transformation written in terms of _____d_e_ as

_ Finite Element Approximation

1,1 (x ,y ) _

(x ,y )

2,2 (x ,y )

1,2

(x ,y )



2,1

1,2 2,2

1,1 2,1

1 1



Figure ____ Bilinear mapping of the canonical square to a quadrilateral_

_ x___ __

y___ __ _

Xi__

Xj__

_ xij

yij _Ni_j___ ___ ______

where Ni_j___ __ is given by _____b__

The transformation is linear on each edge of the element_ In particular_ transforming

the edge _ __ to the physical edge _x___ y__ _ _x___ y__ yields

_ x

y _ _ x__

y__ _ _ _ _

_ _ x__

y__ _ _ _ _

_ __ _ _ _ __

As _ varies from __ to __ x and y vary linearly from _x___ y___ to _x___ y____ The locations

of the vertices ____ and ___ have no e_ect on the transformation_ This ensures that a

continuous approximation in the ___ ___plane will remain continuous when mapped to the

_x_ y__plane_ We have to ensure that the mapping is invertible and we_ll show in Chapter

_ that this is the case when Q is convex_

Problems

__ As noted_ interpolation errors of the biquadratic approximation ______ are the same

order as for a quadratic approximation on a triangle_ Thus_ for example_ the L_

error in interpolating a smooth function u_x_ y_ by a piecewise biquadratic function

U_x_ y_ is O_h___ where h is the length of the longest edge of an element_ The

extra degrees of freedom associated with the cubic and quartic terms do not gen_

erally improve the order of accuracy_ Hence_ we might try to eliminate some shape

functions and reduce the complexity of the approximation_ Unknowns associated

with interior shape functions are only coupled to unknowns on the element and can

easily be eliminated by a variety of techniques_ Considering the biquadratic poly_

nomial in the form _____a__ we might determine c___ so that the coe_cient of the

____ Hierarchical Shape Functions __

quartic term x_y_ vanishes_ Show how this may be done for a _ square canon_

ical element_ Polynomials of this type have been called serendipity by Zienkiewicz

____ In the next section_ we shall see that they are also a part of the hierarchical

family of approximations_ The parameter c___ is said to be _constrained_ since it is

prescribed in advance and not determined as part of the Galerkin procedure_ Plot

or sketch shape functions associated with a vertex and a midside_

___ Hierarchical Shape Functions

We have discussed the advantages of hierarchical bases relative to Lagrangian bases for

one_dimensional problems in Section ___ Similar advantages apply in two and three di_

mensions_ We_ll again use the basis of Szab_o and Babu ska ____ but follow the construction

procedure of Shephard et al_ ___ and Dey et al_ ____ Hierarchical bases of degree p may

be constructed for triangles and squares_ Squares are the simpler of the two_ so let us

handle them _rst_

_____ Hierarchical Shape Functions on Squares

We_ll construct the basis on the canonical element f___ __j _ _ _ __ _ _ _g_ indexing

the vertices_ edges_ and interiors as described for the biquadratic approximation shown

in Figure _____ The hierarchical polynomial of order p has a basis consisting of the

following shape functions_

Vertex shape functions_ The four vertex shape functions are the bilinear functions

_____c_e_

i_j _N

i___ _N

j____ i_ j __ _ ____a_

where

_N _

_ _ _

_ _N

____

_ _ _

_ ____b_

The shape function N_

___ is shown in the upper left portion of Figure ____

Edge shape functions_ For p _ there are _p _ __ shape functions associated with

the midside nodes ___ ___ __ ___ ___ __ and ___ ___

______ __ _N

____ _N

k____ ___a_

______ __ _N

____ _N

k____ ___b_

______ __ _N

____ _N

k____ ___c_

______ __ _N

____ _N

k____ k _ __ _ _ _ _ p_ ___d_

__ Finite Element Approximation

where _N

k____ k _ __ _ _ _ _ p_ are the one_dimensional hierarchical shape functions given

by _____a_ as

k___ rk _ _

Z _

Pk____d_ ___e_

Edge shape functions Nk

___ are shown for k _ __ _ in Figure ____ The edge shape

functions are the product of a linear function of the variable normal to the edge to which

they are associated and a hierarchical polynomial of degree k in a variable on this edge_

The linear function _ _N

j____ _N

j____ j __ _ _blends_ the edge function _ _N

k____ _N

k____

onto the element so as to ensure continuity of the basis_

Interior shape functions_ For p _ there are _p___p____ internal shape functions

associated with the centroid_ Node ___ ___ The _rst internal shape function is the _bubble

function_

N_____

___ __ _ _____ _ ____ ____a_

The remaining shape functions are products of N_____

___ and the Legendre polynomials as

N_____

___ N_____

___ P_____ ____b_

N_____

___ N_____

___ P_____ ____c_

N_____

___ N_____

___ P_____ ____d_

N_____

___ N_____

___ P____P_____ ____e_

N_____

___ N_____

___ P_____ _ _ _ _ ____f_

The superscripts k_ __ and __ resectively_ give the polynomial degree_ the degree of P_____

and the degree of P_____ The _rst six interior bubble shape functions Nk____

___ _ ___ k__

k _ __ __ are shown in Figure ___ These functions vanish on the element boundary

to maintain continuity_

On the canonical element_ the interpolant U___ __ is written as the usual linear com_

bination of shape functions

U___ __

Xi__

Xj__

i_jN_

i_j _

Xk__

Xj__

ck_

_jNk

__j _

Xi__

i__Nk

i___ _

Xk__ X ____k__

ck____

___ Nk____

___ _

____

The notation is somewhat cumbersome but it is explicit_ The _rst summation identi_es

unknowns and shape functions associated with vertices_ The two center summations

identify edge unknowns and shape functions for polynomial orders to p_ And_ the

third summation identi_es the interior unknowns and shape functions of orders to p_

____ Hierarchical Shape Functions __

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Figure ____ Hierarchical vertex and edge shape functions for k _ _upper left__ k

_upper right__ k _ _lower left__ and k _lower right__

Summations are understood to be zero when their initial index exceeds the _nal index_

A degree p approximation has _ _p _ ___ _ _p _ ___p _ ____ unknowns and shape

functions_ where q_ max_q_ ___ This function is listed in Table ___ for p ranging from

_ to __ For large values of p there are O_p__ internal shape functions and O_p_ edge

functions_

_____ Hierarchical Shape Functions on Triangles

We_ll express the hierarchical shape functions for triangular elements in terms of trian_

gular coordinates_ indexing the vertices as __ _ and __ the edges as _ __ and __ and the

centroid as _ _Figure _____ The basis consists of the following shape functions_

Vertex Shape functions_ The three vertex shape functions are the linear barycentric

coordinates _____

i _ __ __ __ i_ i __ _ __ _____

__ Finite Element Approximation

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Figure ___ Hierarchical interior shape functions N_____

___ _ N_____

___ _top__ N_____

___ _ N_____

___ _mid_

dle__ and N_____

___ _ N_____

___ _bottom__

____ Hierarchical Shape Functions __

p Square Triangle

Dimension Dimension

_ _

_ _ __

__ __

_ _ _

_ __ _

_ __ __

_ _ _

Table ____ Dimension of the hierarchical basis of order p on square and triangular

elements_

____

___

____

___

_____

____









1 (1,0,0)

3 (0,0,1)

2 (0,1,0)

Figure ____ Node placement and coordinates for hierarchical approximations on a tri_

angle_

Edge shape functions_ For p there are __p _ __ edge shape functions which are

each nonzero on one edge _to which they are associated_ and vanish on the other two_

Each shape function is selected to match the corresponding edge shape function on a

square element so that a continuous approximation may be obtained on meshes with

both triangular and quadrilateral elements_ Let us construct of the shape functions Nk

_ _

k _ __ _ _ _ _ p_ associated with Edge _ They are required to vanish on Edges _ and _

and must have the form

_ _ __ __ __ _ _ _k____ k _ __ _ _ _ _ p_ ____a_

where _k___ is a shape function to be determined and _ is a coordinate on Edge that

has value __ at Node __ _ at Node _ and _ at Node _ Since Edge is _ __ we have

_ _ __ __ __ _ _ _k____ _ _ _ __

_ Finite Element Approximation

The latter condition follows from _____ with _ __ Along Edge _ _ ranges from _ to

_ and _ ranges from _ to _ as _ ranges from __ to __ thus_ we may select

_ __ _ ____ _ _______ _ __ ____b_

While _ may be de_ned in other ways_ this linear mapping ensures that _ _ _ _ on

Edge _ Compatibility with the edge shape function ____ requires

_ _ __ __ __ _N

k___

__ _ ____ _ __

_k___

where _N

k___ is the one_dimensional hierarchical shape function ___e__ Thus_

_k___

k___

_ _ __ _ ____c_

The result can be written in terms of triangular coordinates by using ____b_ to obtain

_ _ _ __ hence_

_ _ __ __ __ _ _ _k_ _ _ ___ k _ __ _ _ _ _ p_ ____a_

Shape functions along other edges follow by permuting indices_ i_e__

_ _ __ __ __ _ _ _k_ _ _ ___ ____b_

_ _ __ __ __ _ _ _k_ _ _ ___ k _ __ _ _ _ _ p_ ____c_

It might appear that the shape functions _k___ has singularities at _ __ however_ the

one_dimensional hierarchical shape functions have __ _ ___ as a factor_ Thus_ _k___ is a

polynomial of degree k _ _ Using _______ the _rst four of them are

_____ _

p__ _____ _

p____

_____ _r_

____ _ ___ _____ _r_

____ _ ____ _____

Interior shape functions_ The _p _ ___p _ __ internal shape functions for p _ are

products of the bubble function

N_____

_ _ _ ____a_

and Legendre polynomials_ The Legendre polynomials are functions of two of the three

triangular coordinates_ Following Szab_o and Babu ska ____ we present them in terms of

_ _ _ and __ Thus_

N_____

P__ _ _ ___ ____b_

N_____

P__ _ _ ___ ____c_

N_____

P__ _ _ ___ ____d_

N_____

P__ _ _ __P__ _ _ ___ ____e_

N_____

P__ _ _ ___ _ _ _ _ ____f_

____ Three_Dimensional Shape Functions _

The shift in _ ensures that the range of the Legendre polynomials is ____ ___

Like the edge shape functions for a square _____ the edge shape functions for a

triangle _____ are products of a function on the edge __k_ i_ j__ and a function _ i j_ i _

j_ that blends the edge function onto the element_ However_ the edge functions for the

triangle are not the same as those for the square_ The two are related by ____c__ Having

the same edge functions for all element shapes simpli_es construction of the element

sti_ness matrices ____ We can_ of course_ make the edge functions the same by rede_ning

the blending functions_ Thus_ using ____a_c__ the edge function for Edge can be _N

k___

if the blending function is

_ _

_ _ __ _

In a similar manner_ using ___a_ and ____c__ the edge function for the shape function

___ can be _k___ if the blending function is

______ _ ___

Shephard et al_ ___ show that representations in terms of _k involve fewer algebraic

operations and_ hence_ are preferred_

The _rst three edge and interior shape functions are shown in Figure ___ A degree

p hierarchical approximation on a triangle has ____p______p_____p____ unknowns

and shape functions_ This function is listed in Table ____ We see that forp _ __ there are

two fewer shape functions with triangular elements than with squares_ The triangular

element is optimal in the sense of using the minimal number of shape functions for a

complete polynomial of a given degree_ This_ however_ does not mean that the complexity

of solving a given problem is less with triangular elements than with quadrilaterals_ This

issue depends on the partial di_erential equations_ the geometry_ the mesh structure_ and

other factors_

Carnevali et al_ __ introduced shape functions that produce better conditioned ele_

ment sti_ness matrices at higher values of p than the bases presented here ____ Adjerid

et al_ ___ construct an alternate basis that appears to further reduce ill conditioning at

high p_

___ Three_Dimensional Shape Functions

Three_dimensional _nite element shape functions are constructed in the same manner as

in two dimensions_ Common element shapes are tetrahedra and hexahedra and we will

examine some Lagrange and hierarchical approximations on these elements_

Finite Element Approximation

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Figure ___ Hierarchical edge and interior shape functions N_

_ _top left__ N_

_ _top right__

_ _middle left__ N_____

_middle right__ N_____

_bottom left__ N_____

_bottom right__

_____ Lagrangian Shape Functions on Tetrahedra

Let us begin with a linear shape function on a tetrahedron_ We introduce four nodes

numbered _for convenience_ as _ to at the vertices of the element _Figure ______ Im_

posing the usual Lagrangian conditions that Nj_xk_ yk_ zk_ _jk_ j_ k __ _ __ _ gives

____ Three_Dimensional Shape Functions _

the shape functions as

1 (1,0,0,0)

2 (0,1,0,0)

3 (0,0,1,0)

4 (0,0,0,1)

P 1 2 3 4

Figure _____ Node placement for linear shape functions on a tetrahedron and de_nition

of tetrahedral coordinates_

Nj_x_ y_ z_

Dk_l_m_x_ y_ z_

Cj_k_l_m

_ _j_ k_ l_m_ a permutation of __ _ __ _ _____a_

where

Dk_l_m_x_ y_ z_ det

____

_ x y z

_ xk yk zk

_ xl yl zl

_ xm ym zm

____

_ _____b_

Cj_k_l_m det

____

_ xj yj zj

_ xk yk zk

_ xl yl zl

_ xm ym zm

____

_ _____c_

Placing nodes at the vertices produces a linear shape function on each face that is uniquely

determined by its values at the three vertices on the face_ This guarantees continuity of

bases constructed from the shape functions_ The restriction of U to element e is

U_x_ y_ z_

Xj__

cjNj_x_ y_ z__ _____

As in two dimensions_ we may construct higher_order polynomial interpolants by

either mapping to a canonical element or by introducing _tetrahedral coordinates__ Fo_

cusing on the latter approach_ let

j Nj_x_ y_ z__ j __ _ __ _ _____a_

Finite Element Approximation

____

4 (0,0,1)

1 (0,0,0)

3 (0,1,0)

2 (1,0,0)







Figure ____ Transformation of an arbitrary tetrahedron to a right_ unit canonical tetra_

hedron_

and regard j_ j __ _ __ _ as forming a redundant coordinate system on a tetrahedron_

The coordinates of a point P located at _ __ __ __ __ are _Figure _____

VP___

V____

_ _

VP___

V____

_ _

VP___

V____

_ _

VP___

V____

_ _____b_

where Vijkl is the volume of the tetrahedron with vertices at i_ j_ k_ and l_ Hence_ the

coordinates of Vertex _ are ___ __ __ ___ those of Vertex are ___ __ __ ___ etc_ The plane

_ is the plane A___ opposite to vertex __ etc_ The transformation from physical to

tetrahedral coordinates is

____

x_ x_ x_ x_

y_ y_ y_ y_

z_ z_ z_ z_

_ _ _ _

____

_ _____

The coordinate system is redundant as expressed by the last equation_

The transformation of an arbitrary tetrahedron to a right_ unit canonical tetrahedron

_Figure ____ follows the same lines_ and we may de_ne it as

_ N__x_ y_ z__ _ N__x_ y_ z__ N__x_ y_ z__ ______

The face A___ _Figure ____ is mapped to the plane _ __ the face A___ is mapped to

_ __ and A___ is mapped to __ In analogy with the two_dimensional situation_ this

transformation is really the same as the mapping ______ to tetrahedral coordinates_

A complete polynomial of degree p in three dimensions has

_p _ ___p _ __p_ __

______

____ Three_Dimensional Shape Functions _

monomial terms _cf__ e_g__ Brenner and Scott ____ Section _____ With p _ we have

n_ __ monomial terms and we can determine Lagrangian shape functions by placing

nodes at the four vertices and at the midpoints of the six edges _Figure ______ With

p __ we have n_ _ and we can specify shape functions by placing a node at each of

the four vertices_ two nodes on each of the six edges_ and one node on each of the four

faces _Figure ______ Higher degree polynomials also have nodes in the element_s interior_

In general there is _ node at each vertex_ p__ nodes on each edge_ _p____p___ nodes

on each face_ and _p _ ___p _ __p _ ____ nodes in the interior_

_ _

_ __

Figure _____ Node placement for quadratic _left_ and cubic _right_ interpolants on tetra_

hedra_

Example ______ The quadratic shape function N_

_ associated with vertex Node _ of a

tetrahedron _Figure _____ left_ is required to vanish at all nodes but Node __ The plane

_ _ passes through face A___ and_ hence_ Nodes _ __ _ __ __ ___ Likewise_ the plane

_ __ passes through Nodes __ _ _not shown__ and __ Thus_ N_

_ must have the form

_ _ __ __ __ __ _ __ _ _ ____

Since N_

_ _ at Node _ _ _ ___ we _nd _ and

_ _ __ __ __ __ __ _ _ ____

Similarly_ the shape function N_

_ associated with edge Node _ _Figure _____ left_ is

required to vanish on the planes _ _ _Nodes _ __ _ __ __ ___ and _ _ _Nodes __ __

_ __ __ ___ and have unit value at Node _ _ _ _ ____ Thus_ it must be

_ _ __ __ __ __ _ __

_ Finite Element Approximation

__ _

_ _

________

_________

2,2,2

1,2,1

2,1,1 2,2,1

2,1,2

1,2,2



1,1,1

1,1,2





Figure ____ Node placement for a trilinear _left_ and tri_quadratic _right_ polynomial

interpolants on a cube_

_____ Lagrangian Shape Functions on Cubes

In order to construct a trilinear approximation on the canonical cube f__ __ j _ _ _

__ __ _ _g_ we place eight nodes numbered _i_ j_ k__ i_ j_ k __ _ at its vertices _Figure

_____ The shape function associated with Node _i_ j_ k_ is taken as

Ni_j_k___ __ _ _N

i___ _N

j___ _N

k_ _ _____a_

where _N

i____ i __ _ are the hat function _____d_e__ The restriction of U to this element

has the form

U___ __ _

Xi__

Xj__

Xk__

ci_j_kNi_j_k___ __ __ _____b_

Once again_ ci_j_k Ui_j_k U__i_ _j_ k__

The placement of nodes at the vertices produces bilinear shape functions on each

face of the cube that are uniquely determined by values at their four vertices on that

face_ Once again_ this ensures that shape functions and U are C_ functions on a uniform

grid of cubes or rectangular parallelepipeds_ Since each shape function is the product of

one_dimensional linear polynomials_ the interpolant is a trilinear function of the form

U___ __ _ a_ _ a__ _ a__ _ a_ _ a___ _ a__ _ a _ _ a__ _

Other approximations and transformations follow their two_dimensional counterparts_

For example_ tri_quadratic shape functions on the canonical cube are constructed by

placing _ nodes at the vertices_ midsides_ midfaces_ and centroid of the element _Figure

_____ The shape function associated with Node _i_ j_ k_ is given by _____a_ with _N

i___

given by _____b_d__

____ Three_Dimensional Shape Functions _

_____ Hierarchical Approximations

As with the two_dimensional hierarchical approximations described in Section __ we use

Szab_o and Babu ska_s ___ shape function with the representation of Shephard et al_ ____

The basis for a tetrahedral or a canonical cube begins with the vertex functions ______

or _______ respectively_ As noted in Section __ higher_order shape functions are written

as products

i _x_ y_ z_ _k___ __ __i___ __ _ ______

of an entity function _k and a blending function _i_

_ The entity function is de_ned on a mesh entity _vertex_ edge_ face_ or element_ and

varies with the degree k of the approximation_ It does not depend on the shapes

of higher_dimensional entities_

_ The blending function distributes the entity function over higher_dimensional enti_

ties_ It depends on the shapes of the higher_dimensional entities but not on k_

The entity functions that are used to construct shape functions for cubic and tetra_

hedral elements follow_

Edge functions for both cubes and tetrahedra are given by ____c_ and ___e_ as

_k___ p_k _ __

_ _ __ Z _

Pk____d_ k _ _____a_

where _ _ ____ __ is a coordinate on the edge_ The _rst four edge functions are presented

in ______

Face functions for squares are given by _____ divided by the square face blending

function ____a_

_k_______ __ P____P_____ _ _ _ k _ _ k _ _____b_

Here_ ___ __ are canonical coordinates on the face_ The _rst six square face functions are

______ __ ______ __

______ __ ______

___ _ _

______ ___ ______

___ _ _

Face functions for triangles are given by _____ divided the triangular face blending

function ____a_

_k_____ __ __ __ P__ _ _ __P__ _ _ ___ _ _ _ k _ __ k __ _____c_

_ Finite Element Approximation

As with square faces_ _ __ __ __ form a canonical coordinate system on the face_ The

_rst six triangular face functions are

______ __ ______ _ _ __

______ _ _ __ ______

__ _ _ ___ _ _

______ _ _ _ ___ _ _ ___ ______

__ _ _ ___ _ _

Now_ let_s turn to the blending functions_

The tetrahedral element blending function for an edge is

_ij_ __ __ __ __ i j ______a_

when the edge is directed from Vertex i to Vertex j_ Using either Figure ___ or Figure

____ as references_ we see that the blending function ensures that the shape function

vanishes on the two faces not containing the edge to maintain continuity_ Thus_ if i _

and j _ the blending function for Edge ___ _ _which is marked with a _ on the left of

Figure _____ vanishes on the faces _ _ _Face A____ and _ _ _Face A_____

The blending function for a face is

_ijk_ __ __ __ __ i j k ______b_

when the vertices on the face are i_ j_ and k_ Again_ the blending function ensures that

the shape function vanishes on all faces but Aijk_ Again referring to Figures ___ or

_____ the blending function ____ vanishes when _ _ _Face A_____ _ _ _Face A_____

and _ _ _Face A_____

The cubic element blending function for an edge is more di_cult to write with our

notation_ Instead of writing the general result_ let_s consider an edge parallel to the _

axis_ Then

_____j_k___ __ _

_ _ __

j___ _N

k_ __ ______a_

The factor __ _ ____ adjusts the edge function to ______ as described in the paragraph

following ______ The one_dimensional shape functions _N

j___ and _N

k_ _ ensure that the

shape function vanishes on all faces not containing the edge_ Blending functions for other

edges are obtained by cyclic permutation of __ __ and and the index_ Thus_ referring

to Figure ____ the edge function for the edge connecting vertices _ __ _ and _ __ is

___________ __ _

_ _ __

____ _N

__ __

Since _N

_____ _ _cf_ _____b___ the shape function vanishes on the rear face of the cube

shown in Figure ____ Since _N

____ __ the shape function vanishes on the top face of

____ Three_Dimensional Shape Functions _

the cube of Figure ____ Finally_ the shape function vanishes at _ _ and_ hence_ on

the left and right faces of the cube of Figure ____ Thus_ the blending function ______a_

has ensured that the shape function vanishes on all but the bottom and front faces of

the cube of Figure ____

The cubic face blending function for a face perpendicular to the _ axis is

_i_j_k___ __ _ _N

i_____ _ _____ _ ___ ______b_

Referring to Figure ____ the quadratic terms in _ and ensure that the shape func_

tion vanishes on the right_ left __ ___ top_ and bottom _ __ faces_ The one_

dimensional shape function _N

i___ vanishes on the rear __ ___ face when i _ and on

the front __ __ face when i _ thus_ the shape function vanishes on all faces but the

one to which it is associated_

Finally_ there are elemental shape functions_ For tetrahedra_ there are _p _ ___p _

__p _ ____ elemental functions for p that are given by

Nk______

_ _ __ __ __ __ _ _ _ _P__ _ _ __P__ _ _ __P__ _ _ ___

_ _ _ _ _ _ k _ _ k _ __ _ _ _ _ p_ _____a_

The subscript _ is used to identify the element_s centroid_ The shape functions vanish

on all element faces as indicated by the presence of the multiplier _ _ _ __ We could

also split this function into the product of an elemental function involving the Legendre

polynomials and the blend involving the product of the tetrahedral coordinates_ However_

this is not necessary_

For p _ there are the following elemental shape functions for a cube

Nk______

_ ___ __ _ __ _ _____ _ _____ _ __P____P____P__ __ _ _ _ _ _ _ k _ __

_____b_

Again_ the shape function vanishes on all faces of the element to maintain continuity_

Adding_ we see that there are _p_____p____p______ element modes for a polynomial

of order p_

Shephard et al_ ___ also construct blending functions for pyramids_ wedges_ and prisms_

They display several shape functions and also present entity functions using the basis of

Carnevali et al_ ___

Problems

__ Construct the shape functions associated with a vertex_ an edge_ and a face node

for a cubic Lagrangian interpolant on the tetrahedron shown on the right of Figure

_____ Express your answer in the tetrahedral coordinates _______

__ Finite Element Approximation

__________________________

_________

1 (x ,y )

1 1

1 (0,0) 2 (1,0)

3 (0,1)





2 (x ,y ) 2 2

3 (x ,y ) 3 3



1 

2 3

Figure _____ Nomenclature for a _nite element in the physical _x_ y__plane and for its

mapping to a canonical element in the computational ___ ___plane_

___ Interpolation Error Analysis

We conclude this chapter with a brief discussion of the errors in interpolating a function u

by a piecewise polynomial function U_ This work extends our earlier study in Section __

to multi_dimensional situations_ Two_ and three_dimensional interpolation is_ naturally_

more complex_ In one dimension_ it was su_cient to study limiting processes where mesh

spacings tend to zero_ In two and three dimensions_ we must also ensure that element

shapes cannot be too distorted_ This usually means that elements cannot become too

thin as the mesh is re_ned_ We have been using coordinate mappings to construct

bases_ Concentrating on two_dimensional problems_ the coordinate transformation from

a canonical element in_ say_ the ___ ___plane to an actual element in the _x_ y__plane must

be such that no distorted elements are produced_

Let_s focus on triangular elements and consider a linear mapping of a canonical unit_

right_ __ triangle in the ___ ___plane to an element e in the _x_ y__plane _Figure ______

More complex mappings will be discussed in Chapter __ Using the transformation _____

to triangular coordinates in combination with the de_nitions _____ and _____ of the

canonical variables_ we have

__ x

y_ y_ y_

_ _ _

x_ x_ x_

y_ y_ y_

_ _ _

_ _ _ _ _

_ ______

The Jacobian of this transformation is

Je _ _ x_ x_

y_ y_ __ ____a_

____ Three_Dimensional Shape Functions __

Di_erentiating _______ we _nd the determinant of this Jacobian as

det_Je_ _x_ _ x___y_ _ y__ _ _x_ _ x___y_ _ y___ ____b_

Lemma ______ Let he be the longest edge and _e be the smallest angle of Element e_

then

sin _e _ det_Je_ _ h_

sin _e_ ______

Proof_ Label the vertices of Element e as __ _ and __ their angles as __ _ __ _ ___ and

the lengths of the edges opposite these angles as h__ h__ and h_ _Figure ______ With

__ _e being the smallest angle of Element e_ write the determinant of the Jacobian as

det_Je_ h_h_ sin _e_

Using the law of sines we have h_ _ h_ _ h_ he_ Replacing h_ by h_ in the above

expression yields the right_hand inequality of _______ The triangular inequality gives

h_ _ h_ _ h__ Thus_ at least one edge_ say_ h_ _ h___ This yields the left_hand

inequality of _______

Theorem ______ Let __x_ y_ _ Hs__e_ and !____ __ _ Hs____ be such that __x_ y_

!____ __ where _e is the domain of element e and __ is the domain of the canonical element_

Under the linear transformation ______ _ there exist constants cs and Cs_ independent of

__ !__ he_ and _e such that

cs sins____ _ehs__

e j_js_e _ j!_js__ _ Cs sin____ _ehs__

e j_js_e ____a_

where the Sobolev seminorm is

j_j_

s_e

Xj_j_s

_D____dxdy ____b_

with D_u being a partial derivative of order j_j s _cf_ Section __ _

Proof_ Let us begin with s __ where

__dxdy det_Je_ ZZ

!__d_d_

j_j_

det_Je_j!_j_

__ _

Dividing by det_Je_ and using ______

j_j_

sin _eh_

_ j!_j_

__ _

j_j_

sin _eh_

_ Finite Element Approximation

Taking a square root_ we see that ____a_ is satis_ed with c_ _ and C_ p_

With s __ we use the chain rule to get

_x !___x _ !___x_ _y !___y _ !___y_

Then_

j_j_

___

_ __

y_dxdy det_Je_ ZZ

_g__e!__

_ _ g__e!__ !__ _ g__e!__

__d_d_

where

g__e __

x _ __

y _ g__e _x_x _ _y_y_ g__e __

_ __

Applying the inequality ab _ _a_ _ b___ to the center term on the right yields

j_j_

_e _ det_Je_ ZZ

_g__e

!__

_ _ g__e_!__

_ _ !__

__ _ g__e

!__

__d_d__

Letting

_ max_jg__e _ g__ej_ jg__e _ g__ej_

and using ____b__ we have

j_j_

_e _ det_Je__j!_j_

__ _ _____a_

Either by using the chain rule above with _ x and y or by inverting the mapping

_______ we may show that

det_Je_

_ _y _

det_Je_

_ _x _

det_Je_

_ _y _

det_Je_

From ______ jx_j_ jx_j_ jy_j_ jy_j _ he_ thus_ using _______ we have j_xj_ j_yj_ j_xj_ j_yj _

__he sin _e__ Hence_

_ _

_he sin _e__ _

Using this result and ______ with _____a__ we _nd

j_j_

_e _

sin _e j!_j_

__ _ _____b_

Hence_ the left_hand inequality of ____a_ is established with c_ ___

To establish the right inequality_ we invert the transformation and proceed from __

to _e to obtain

j!_j_

__ _

!_j_j_

det_Je_

_____a_

____ Three_Dimensional Shape Functions __

with

!_ max_j!g__e _ !g__ej_ j!g__e _ !g__ej__

!g__e x_

_ x_

_ !g__e x_y_ _ x_y__ !g__e y_

_ _ y_

We_ve indicated that jx_j_ jx_j_ jy_j_ jy_j _ he_ Thus_ !_ _ h_

and_ using _______ we _nd

j!_j_

__ _

sin _e j_j_

_ _____b_

Thus_ the right inequality of ____b_ is established with C_ p_

The remainder of the proof follows the same lines and is described in Axelsson and

Barker ___

With Theorem ____ established_ we can concentrate on estimating interpolation errors

on the canonical triangle_ For simplicity_ we_ll use the Lagrange interpolating polynomial

___ __

Xj__

!u__j_ _j_Nj___ ___ ______

with n being the number of nodes on the standard triangle_ However_ with minor alter_

ations_ the results apply to other bases and_ indeed_ other element shapes_ We proceed

with one preliminary theorem and then present the main result_

Theorem ______ Let p be the largest integer for which the interpolant ______ is exact

when !u___ __ is a polynomial of degree p_ Then_ there exists a constant C _ _ such that

j!u _ ! Ujs__ _ Cj!ujp____ _ _u _ Hp_______ s __ __ _ _ _ _ p _ __ ______

Proof_ The proof utilizes the Bramble_Hilbert Lemma and is presented in Axelsson and

Barker ___

Theorem ______ Let _ be a polygonal domain that has been discretized into a net of

triangular elements _e_ e __ _ _ _ __N__ Let h and _ denote the largest element edge

and smallest angle in the mesh_ respectively_ Let p be the largest integer for which ______

is exact when !u___ __ is a complete polynomial of degree p_ Then_ there exists a constant

C _ __ independent of u _ Hp__ and the mesh_ such that

ju _ Ujs _

Chp___s

_sin __s jujp___ _u _ Hp______ s __ __ ______

Remark __ The results are restricted s __ _ because_ typically_ U _ H_ _ Hp___

_ Finite Element Approximation

Proof_ Consider an element e and use the left inequality of ____a_ with _ replaced by

u _ U to obtain

ju _ Uj_

s_e _ c__

s sin__s__ _eh__s__

e j!u _ !U j_

s__ _

Next_ use ______

ju _ Uj_

s_e _ c__

s sin__s__ _eh__s__

e Cj!uj_

p____ _

Finally_ use the right inequality of ____a_ to obtain

ju _ Uj_

s_e _ c__

s sin__s__ _eh__s__

e CC_

p__ sin__ _eh_p

e juj_

p___e

Combining the constants

ju _ Uj_

s_e _ C sin__s _eh__p___s_

e juj_

p___e

Summing over the elements and taking a square root gives _______

A similar result for rectangles follows_

Theorem ______ Let the rectangular domain _ be discretized into a mesh of rectangular

elements _e_ e __ _ _ _ __N__ Let h and _ denote the largest element edge and smallest

edge ratio in the mesh_ respectively_ Let p be the largest integer for which ______ is exact

when !u___ __ is a complete polynomial of degree p_ Then_ there exists a constant C _ __

independent of u _ Hp__ and the mesh_ such that

ju _ Ujs _

Chp___s

_s jujp___ _u _ Hp______ s __ __ _______

Proof_ The proof follows the lines of Theorem ____ ___

Thus_ small and large _near __ angles in triangular meshes and small aspect ratios

_the minimum to maximum edge ratio of an element_ _ in a rectangular mesh must be

avoided_ If these quantities remain bounded then the mesh is uniform as expressed by

the following de_nition_

De_nition ______ A family of _nite element meshes "h is uniform if all angles of all

elements are bounded away from _ and _ and all aspect ratios are bounded away from

zero as the element size h _ __

With such uniform meshes_ we can combine Theorems ____ _____ and ___ to obtain

a result that appears more widely in the literature_

Theorem ______ Let a family of meshes "h be uniform and let the polynomial inter_

polant U of u _ Hp__ be exact whenever u is a complete polynomial of degree p_ Then

there exists a constant C _ _ such that

ju _ Ujs _ Chp___sjujp___ s __ __ _______

____ Three_Dimensional Shape Functions __

Proof_ Use the bounds on _ and _ with ______ and _______ to rede_ne the constant C

and obtain ________

Theorems ___ _ ____ only apply when u _ Hp___ If u has a singularity and belongs

to Hq___ q _ p_ then the convergence rate is reduced to

ju _ Ujs _ Chq___sjujq___ s __ __ ______

Thus_ there appears to be little bene_t to using p th_degree piecewise_polynomial inter_

polants in this case_ However_ in some cases_ highly graded nonuniform meshes can be

created to restore a higher convergence rate_

__ Finite Element Approximation

Bibliography

___ S_ Adjerid_ M_ Ai_a_ and J_E_ Flaherty_ Hierarchical _nite element bases for triangular

and tetrahedral elements_ Computer Methods in Applied Mechanics and Engineering_

____ to appear_

__ O_ Axelsson and V_A_ Barker_ Finite Element Solution of Boundary Value Problems_

Academic Press_ Orlando_ ____

___ S_C_ Brenner and L_R_ Scott_ The Mathematical Theory of Finite Element Methods_

Springer_Verlag_ New York_ ____

__ P_ Carnevali_ R_V_ Morric_ Y_Tsuji_ and B_ Taylor_ New basis functions and com_

putational procedures for p_version _nite element analysis_ International Journal of

Numerical Methods in Enginneering_ _______#_____ _____

___ S_ Dey_ M_S_ Shephard_ and J_E_ Flaherty_ Geometry_based issues associated with

p_version _nite element computations_ Computer Methods in Applied Mechanics and

Engineering_ ______ # ___ _____

___ M_S_ Shephard_ S_ Dey_ and J_E_ Flaherty_ A straightforward structure to construct

shape functions for variable p_order meshes_ Computer Methods in Applied Mechanics

and Engineering_ _____#___ _____

___ B_ Szab_o and I_ Babu ska_ Finite Element Analysis_ John Wiley and Sons_ New York_

_____

___ O_C_ Zienkiewicz_ The Finite Element Method_ McGraw_Hill_ New York_ third edition_

_____

Chapter _

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