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Workbook of Applications in VectorSpace C++ Library 453

Mixed and Hybrid Finite Element Methods

Eq. 512

where and with over-bar are fixed nodal flux boundary conditions and fixed nodal temperature boundary

conditions, respectively, while h in Eq. 512 can be specified as a function on an element boundary. The fixed

nodal boundary conditions, and , in Eq. 511 and Eq. 512 are encountered frequently in finite element

method and are taken care of by fe.lib as default behaviors behind the scene. This is consistent with the treat-

ment of the result of the displacementboundary conditions as reaction, and substrates out of the nodal force term

as -= K .

By inspecting on Eq. 59 and Eq. 510, the derivatives of temperature field exist. C0-continuity of T on the

element interior and boundaries is required. This is needed to guarantees that Eq. 510 over the entire problemdomain is integrable. Otherwise, the integration goes to infinity at the discontinuities. Two triangular elements

are considered in the following computations. Figure 51a&b shows that two triangular elements with linear tem-

perature field with three corner nodes and either a constant heat flux with one node at the center of the element or

linear heat flux with three nodes at the Gaussian integration points. We notice that the temperature nodes on the

three corners is necessary to ensure the C0-continuity on the element boundaries, while the heat flux, with no

derivatives of heat flux in Eq. 59 and Eq. 510, can be discontinuous at the element boundaries. Therefore, the

constant heat flux element has one node at the center of the element (Figure 51a), and the linear heat flux ele-

ment has three nodes at the Gaussian integration points (Figure 51b). Actually, requiring heat flux to be C0-con-tinuity on the element boundaries, had we use three corner nodes for the heat flux, may cause physically

incorrect conditions.1

Recall in Section 4.2.5, the mixed formulation for one dimensional beam bending problem, we have extended

fe.lib with the object-oriented modeling for developing matrix substructuring to solve system of equations in

submatrices similar to Eq. 58.

1. e.g. as discussed in p.327 in Zienkiewicz, O.C. and R.L. Taylor, 1991, The finite element method, 4th ed., vol. 1.McGraw-Hill, Inc., UK.

f2 T f de

T h dh

e

Ch he+=

h g

h g

f u

Figure 51 Triangluar elements with linear temperature with three corner nodes, and eitherconstant heat flux with one node at the center or linear heat flux with three nodes at three Gaussintegration points.

T-nodesq-nodes

(a) Constantheat flux

(b) Linearheat flux

T = 0oC

T = 30oC

q = 0q = 0

(c)
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Advanced Finite Element Methods

454 Workbook of Applications in VectorSpace C++ Library

Chapter 5

The Program Listing 51 implements the constant heat flux triangular element for the mixed formulation. For

global discretization {h, qh}, with its element discretization as qe, the heat flux is constant over entire domain

ofqe. Only one node at the center of the element is needed. However we still need to specify the three corner

nodes in order to (1) compute the coordinates of the center node, (2) define coordinate transformation rule, and(3) perform integration. Therefore, the three additional corner nodes of q

e are defined as geometrical nodes.

The geometrical nodes has no variable associated with them. A simple trick using fe.lib is to disable all geo-

metrical nodes by specifying all these nodes to have Dirichlet type boundary condition, so they will be left out of

the global stiffness matrix. The A-submatrix are defined by

1 double k_x = 1.0, k_y = 1.0,

2 k_inv[2][2] = { {1.0/k_x, 0.0 }, // , for isotropy

3 {0.0, 1.0/k_y }};4 C0 K_inv = MATRIX("int, int, const double*", 2, 2, k_inv[0]);

5 Heat_Mixed_Formulation::Heat_Mixed_Formulation(int en, Global_Discretization& gd) :

6 Element_Formulation_Couple(en, gd) {

7 Quadrature qp(2, 4);

8 H1 L(2, (double*)0, qp),

9 n = INTEGRABLE_VECTOR_OF_TANGENT_BUNDLE("int, int, Quadrature", 3, 2, qp),

10 L0 = L[0], L1 = L[1], L2 = 1.0 - L0 - L1; // area coordinates for a triangle

11 n[0] = L0; n[1] = L1; n[2] = L2;12 C0 x = MATRIX("int, int, C0&, int, int", 3, 2, xl, 0, 0);

13 H1 X = n*x;

14 J dv(d(X).det()/2.0);

15 H0 N = INTEGRABLE_VECTOR("int, Quadrature", 4, qp);

16 N[0] = N[1] = N[2] = 0.0; N[3] = 1.0;

17 H0 N_q = ((~N) || C0(0.0)) &

18 (C0(0.0) || (~N) ); //

19 stiff &= ((~N_q) * (K_inv * N_q)) | dv;20 }

The area coordinates for triangle,L0,L1, andL2, are used for coordinate transformation and integration. The first

three nodes are geometrical nodes at three corners and the fourth-node is the q-node at the center. A reference

matrix x (line 12) is constructed to refer to the first three coordinates of xl, which leaves out the q-node. To

compute the Jacobian we notice that there is a factor of 1/2 for a triangular element comparing to the a quadrilat-

eral element where the factor = 1 (note that a factor of 1/6 is to be used for a 3-D tetrahedra element). The q-

shape function, N (line 16), is defined that the first three shape functions corresponding to three geometrical

nodes are zero; i.e., N[0] = N[1] = N[2] = 0. And the fourth shape function is one; i.e., N[3] = 1. The ele-

ment stiffness matrix so constructed will have the size of 8 8, with only the 2 2 submatrix at the lower-right

corner, corresponding to the center q-node, contains no zero components. All the geometrical nodes are to be

specified with Dirichlet type boundary condition. Dummy variables corresponding to these geometrical nodes

will all be eliminated from the global matrix and the global vector. Therefore, the no trivial 2 2 element subma-

trix will enter the global stiffness matrix.

The C-submatrix can be defined as

1 1 ij 1 0

0 1

= =

A q 1 q( ) de

=


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Chapter 5

hdisable all geometrical nodes on q

gtop boundary: T = 30o

bottom boundary: T = 0o

for diagonal submatrix A

for off-diagonal submatrix C

A-submatrix element formulation

area coordinates for triangle

linear triangular shape function for

coordinate transformation

= 1.0 (constant over element)

N[0], N[1], N[2] are geometrical nodes

N[3] is constant over element domain

q

ena[0] = first_corner_node_no; ena[1] = first_corner_node_no+1+row_t_node_no;

ena[2] = first_corner_node_no+row_t_node_no;

elem = new Omega_eh(element_no+1, 0, 0, 3, ena); omega_eh_array().add(elem);

}

}}

gh_on_Gamma_h_i::gh_on_Gamma_h_i(int i, int df, Omega_h& omega_h) : gh_on_Gamma_h() {

gh_on_Gamma_h::__initialization(df, omega_h);

if(i == 0) {

for(int j = (row_t_node_no-1)*6*(row_t_node_no-1);

j < (row_t_node_no*row_t_node_no+(row_t_node_no-1)*6*(row_t_node_no-1)); j++)

for(int k = 0; k < 2; k++) {

the_gh_array[node_order(j)](k) = gh_on_Gamma_h::Dirichlet;

the_gh_array[node_order(j)][k] = 0.0;

}

} else if(i == 1) {

for(int j = 0; j < row_t_node_no; j++) {

the_gh_array[node_order(row_t_node_no*(row_t_node_no-1)+j)](0) =

gh_on_Gamma_h::Dirichlet;

the_gh_array[node_order(row_t_node_no*(row_t_node_no-1)+j)][0] =

((double)(row_t_node_no-1))*10.0;

the_gh_array[node_order(j)](0) = gh_on_Gamma_h::Dirichlet;

the_gh_array[node_order(j)][0] = 0.0;

}

}

}static const int q_ndf = 2; static Omega_h_i oh_q(0);

static gh_on_Gamma_h_i q_gh(0, q_ndf, oh_q); static U_h q_h(q_ndf, oh_q);

static Global_Discretization q_gd(oh_q, q_gh, q_h);

static const int T_ndf = 1; static Omega_h_i oh_T(1);

static gh_on_Gamma_h_i T_gh(1, T_ndf, oh_T); static U_h T_h(T_ndf, oh_T);

static Global_Discretization T_gd(oh_T, T_gh, T_h);

static Global_Discretization_Couple gdc(T_gd, q_gd);

class Heat_Mixed_Formulation : public Element_Formulation_Couple {

public:

Heat_Mixed_Formulation(Element_Type_Register a) : Element_Formulation_Couple(a) {}Element_Formulation *make(int, Global_Discretization&);

Heat_Mixed_Formulation(int, Global_Discretization&);

Element_Formulation_Couple *make(int, Global_Discretization_Couple&);

Heat_Mixed_Formulation(int, Global_Discretization_Couple&);

};

Element_Formulation* Heat_Mixed_Formulation::make(int en, Global_Discretization& gd) {

return new Heat_Mixed_Formulation(en,gd);

}

Heat_Mixed_Formulation::Heat_Mixed_Formulation(int en, Global_Discretization& gd) :

Element_Formulation_Couple(en, gd) {Quadrature qp(2, 4);

H1 L(2, (double*)0, qp),

n = INTEGRABLE_VECTOR_OF_TANGENT_BUNDLE("int, int, Quadrature", 3, 2, qp),

L0 = L[0], L1 = L[1], L2 = 1.0 - L0 - L1;

n[0] = L0; n[1] = L1; n[2] = L2;

C0 x = MATRIX("int, int, C0&, int, int", 3, 2, xl, 0, 0);

H1 X = n*x;

J dv(d(X).det()/2.0);

H0 N = INTEGRABLE_VECTOR("int, Quadrature", 4, qp);

N[0] = N[1] = N[2] = 0.0;

N[3] = 1.0;


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Chapter 5

1 Heat_Mixed_Formulation::Heat_Mixed_Formulation(int en, Global_Discretization_Couple& gdc)

2 : Element_Formulation_Couple(en, gdc) {

3 Quadrature qp(2, 4);

4 H1 L(2, (double*)0, qp),5 n = INTEGRABLE_VECTOR_OF_TANGENT_BUNDLE(

6 "int, int, Quadrature", 3/*nen*/, 2/*nsd*/, qp),

7 L0 = L[0], L1 = L[1], L2 = 1.0 - L0 - L1;

8 n[0] = L0; n[1] = L1; n[2] = L2;

9 H1 X = n*xl;

10 H0 nx = d(n) * d(X).inverse();

11 J dv(d(X).det()/2.0);

12 H0 N = INTEGRABLE_VECTOR("int, Quadrature", 4/*nen*/, qp);13 N[0] = N[1] = N[2] = 0.0; N[3] = 1.0;

14 H0 N_q = ( (~N) || C0(0.0)) &

15 (C0(0.0) || (~N)); //

16 stiff &= (nx * N_q) | dv;

17 }

The element stiffness matrix so generated has the size of 3 8. The three rows (number of equations) corre-

sponding to three temperature nodes at the corner. After the element to global mapping, the first 6 columns (cor-responding to the number of dummy variables) for the geometrical nodes will not enter the global stiffness

matrix. Only the last 2 columns (corresponding to the number of variables) for the center q-node will survive.

After the submatrices have been formed, the solution of system of equation in Eq. 58 is the subject ofcon-

straint optimization problem (see Section 2.3.3 in Chapter 2). In the context of finite element problem, the objec-

tive functional for optimization is quadratic. The problem is further restricted to a quadratic programming

problem, in which only one step along the search path is needed to reach the exact solution (see introduction and

its example in page 145). We discuss the range space methodand null space method(see page 149) to solve Eq.58 in the followings.

From first equation ofEq. 58 we have

A + CT = f1 Eq. 513

Considering A is symmetrical positive definitive, therefore it can be inverted, we can solve for q by

= A-1 f1 - A-1CT Eq. 514

Substituting Eq. 514 into second equation ofEq. 58,C = f2, we have

CA-1 f1 - C A-1CT = f2 Eq. 515

Therefore, can also be solved considering that the similarity transformation ofA-1 as C A-1CT preserves

the symmetrical positive definitiveness ofA-1. An alternative view is that C A-1CT is the projection of inverse

C T q de

=

q T

q T

q

T

T
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Chapter 5

17 for(int i = 0; i < q_h.total_node_no(); i++)

18 cout


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Both the range space and null space method reproduce the exact solution up to round-off error for the current

problem. The nodal temperature solutions at the second row is = 10 oC, and the third row is = 20 oC. All

heat flux center nodes have the values of = [0, -10]T per unit length. The default method in Program Listing

51 is the range space method. The null space method can be activated by setting macro definition__NULL_SPACE_METHOD at compile time.

The linear heat flux with three nodes at the Gaussian integration points (see Figure 51b) can be activated by

setting macro definition __TEST_THREE_NODES_DISCONTINUOUS_HEAT_FLUX at compile time. The

triangular element in this implementation used shape functions degenerated from bilinear 4-node element. The

results of this linear heat flux element is identical to that of the constant heat flux element. Considering that T-

field only for a 3-node triangular element vary linearly. The heat flux, computed from the Fouries law, is the

derivative of the linear T-field, which is constant. Therefore, the linear heat flux element does not give any

improvement in the accuracy of the solution comparing to the constant heat flux element. Actually, the linear

heat flux element produces the same result as the constant heat flux element. This is known as the limitationprin-

ciple by Fraeijs de Veubeke.

With the assistance of object-oriented modeling provided in fe.lib for handling the matrix substructuring, it

may seems to be a piece of cake to implement the multiple-field mixed formulation. It is not so for most existing

finite element programs. It has been remarked that the full-scale multiple-field formulations are rarely imple-

mented for practical computation.1

The discontinuous heat flux field, with nodes reside only inside an element, not only avoid physically incor-

rect conditions as mentioned earlier, but it also has the advantage that the heat flux field can be eliminated at the

element level. Every heat flux node belongs only to its containing element; i.e., there are no shared heat flux

nodes with the neighboring elements. Therefore, from Eq. 516 that

= (C A-1CT)-1 (CA-1 f1 - f2)

We can redefine element stiffness matrix and element force vector as

ke = Ce Ae-1CeT, and fe= CeAe-1 f1e - f2

e Eq. 519

respectively, with

Eq. 520

and,

Eq. 521

1. see p.206 in Hughes, T. J.R., The finite element method: linear static and dynamic finite element analysis, Prentice-Hall,Inc., Englewood Cliffs, New Jersey.

T T

q

T

f2e T f d

e=

T h dhe Ceh he 0=


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Chapter 5

Since in this discontinuous heat flux element no heat flux boundary condition can be specified, the temperature

boundary conditions in Eq. 58 are simply

f1 = - Eq. 522

Substituting the element level vector ofEq. 522 into the second equation in Eq. 519, we have

fe = - CeAe-1 - f2e= - Ce Ae-1 - f2

e

= -ke - f2e Eq. 523

The first term in the right-hand-side is consistent with standard implementation of finite element on the essential

boundary conditions. The mixed form ofEq. 519 can be easily implemented which does not require the mecha-

nism provided in the fe.lib to express matrix substructuring. Program Listing 52 implements Eq. 519 (project

mixed_T_heat_conduction in project workspace file fe.dsw). The temperature solutions of the computation

is certainly identical to the full-fledged mixed formulation. The program is significant simplified comparing to

the full-scale mixed formulation.

CT

g g

CeT

gege

CeT

gege

ge ge


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#include "include\fe.h"

static row_node_no = 4;

EP::element_pattern EP::ep = EP::SLASH_TRIANGLES;

Omega_h::Omega_h() {double coord[4][2] = {{0.0, 0.0}, {3.0, 0.0}, {3.0, 3.0}, {0.0, 3.0}};

int control_node_flag[4] = {TRUE, TRUE, TRUE, TRUE};

block(this, row_node_no, row_node_no, 4, control_node_flag, coord[0]);

}

gh_on_Gamma_h::gh_on_Gamma_h(int df, Omega_h& omega_h) { __initialization(df, omega_h);

for(int j = 0; j < row_node_no; j++) {

the_gh_array[node_order(row_node_no*(row_node_no-1)+j)](0) =

the_gh_array[node_order(j)](0) = gh_on_Gamma_h::Dirichlet;

the_gh_array[node_order(row_node_no*(row_node_no-1)+j)][0] =

((double)(row_node_no-1))*10.0;

the_gh_array[node_order(j)][0] = 0.0;

}

}

class HeatMixedT3 : public Element_Formulation {

public:

HeatMixedT3(Element_Type_Register a) : Element_Formulation(a) {}

Element_Formulation *make(int, Global_Discretization&);

HeatMixedT3(int, Global_Discretization&);

};

Element_Formulation* HeatMixedT3::make(int en, Global_Discretization& gd) {

return new HeatMixedT3(en,gd); }double k_x = 1.0, k_y = 1.0, k_inv[2][2] = { {1.0/k_x, 0.0}, { 0.0, 1.0/k_y}};

C0 K_inv = MATRIX("int, int, const double*", 2, 2, k_inv[0]);

HeatMixedT3::HeatMixedT3(int en, Global_Discretization& gd) : Element_Formulation(en, gd) {

Quadrature qp(2, 4);

H1 L(2, (double*)0, qp),

n = INTEGRABLE_VECTOR_OF_TANGENT_BUNDLE("int, int, Quadrature", 3, 2, qp),

L0 = L[0], L1 = L[1], L2 = 1.0 - L0 - L1;

n[0] = L0; n[1] = L1; n[2] = L2;

H1 X = n*xl; H0 nx = d(n) * d(X).inverse(); J dv(d(X).det()/2.0);

H0 N = INTEGRABLE_SCALAR("Quadrature", qp); N = 1.0;

H0 N_q = ((~N) || C0(0.0)) &(C0(0.0)|| (~N));

C0 C = (nx * N_q) | dv,

A = ((~N_q) * (K_inv * N_q)) | dv,

A_inv = A.inverse();

stiff &= C*A_inv*(~C);

}

Element_Formulation* Element_Formulation::type_list = 0;

Element_Type_Register element_type_register_instance;

static HeatMixedT3 heatmixedt3_instance(element_type_register_instance);

int main() {int ndf = 1; Omega_h oh; gh_on_Gamma_h gh(ndf, oh); U_h uh(ndf, oh);

Global_Discretization gd(oh, gh, uh);

Matrix_Representation mr(gd);

mr.assembly();

C0 u = ((C0)(mr.rhs())) / ((C0)(mr.lhs()));

uh = u; uh = gh;

cout


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Chapter 5

Hourglass Element

For the irreducible formulation in Chapter 4, 2 2 Gaussian integration points are used to compute element

stiffness matrix for bilinear 4-node element.

Eq. 524

where

Eq. 525

and a = 0, 1, 2, 3. A reduced integration (1 Gauss point at the center of the element) will result in rank deficiency

of the element stiffness matrix. The rank of the element stiffness matrix is the number of integration points times

the number of independent relations. In the case of heat conduction, the number of independent relations is the

number of equations relating the heat flux [qx, qy] and the temperature gradients [T/x, T/y] by Fourier lawof heat conduction in 2-D. Therefore, the rank of the stiffness matrix for 1-point Gauss integration rule is 2 (=2 1). For the element stiffness matrix of size 4 4 in Eq. 524, the 1-point Gauss integration leads to rank defi-

ciency of 2 (= 4-2). We can consider that the 1-point integration stiffness matrix is span bybx = {bxa,bya} in Eq.

525, which are bases in 4. The two spurious zero energy modes in the null space are constant solution mode

with nodal solution ofsa = [1, 1, 1, 1] and hourglass mode with nodal solution of ha = [-1, 1, -1, 1]. The hour-

glass mode is illustrated in Figure 52. The two zero energy modes are orthogonal tobx

, and Eq. 526

where we also denote x0 = x, and x1 = y. We notice that for isoparametric coordinate transformation xi =Naxia ,

we have the relation

ke N( )TN( )d

e

BTB( )de

= =

Ba bxa

bya

Na

x----------

Nay

----------

-1

-0.5

0

0.5

1

-1

-0.5

0

0.5

1

-1

-0.5

0

0.5

1

-1

-0.5

0

0.5

-1

-0.5

0

0.5

-1

-0.5

0

.5

1

Figure 52 The hourglass mode for heat conduction problem.

T

bxis 0= bxi

h 0=


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Eq. 527

The constant nodal solution sa is considered proper, since a constant temperature field produces no heat flux isas expected, while the hourglass mode ha has gradient far from zero (see Figure 52) but no heat flux production,

which is considered improper. Therefore, we expect the 4-node element stiffness to produce a rank 3 matrix.

The strategy is to use a so-called trial hourglass mode, a , to construct a correct-ranked stiffness ke in a way thatis very economical to compute as

ke = ke(1-point) + ke(hourglass) Eq. 528

The first term on the right hand side, ke(1-point) is standard stiffness matrix computed with only one Gaussianintegration point. The second term ke(hourglass) is to be constructed with the trial hourglass mode a. The generalform of the trial hourglass mode, a , in

4 is span by the four bases [bxa,bya, sa, ha] as

a = a0bxa + a1bya + a2sa+ ha Eq. 529

a is required to be orthogonal with an arbitrary lineartemperature field of Ta

Ta = c0 xa + c1ya + c2sa Eq. 530

For a to be always orthogonal with arbitrary coefficients, ci , in Eq. 530, we have

a xa = 0 , a ya = 0, and asa = 0 Eq. 531

From last part ofEq. 531, no component ofa is in sa, so in Eq. 529 a2 = 0, and we have this equation re-writ-ten as

a = a0bxa + a1bya + ha Eq. 532

Substituting Eq. 532 into a xa = 0 in the first part ofEq. 531, and usingbxa xa = 1, andbya xa = 0 as indicatedin Eq. 527, we have

a0 = - ha xa Eq. 533

Similarly, substituting Eq. 532 into a ya = 0 in the second part ofEq. 531, and usingbya ya = 1, andbxa ya = 0

as in Eq. 527, we have

a1 = - ha ya Eq. 534

Therefore, we show that

a = ha - (ha xa)bxa - (ha ya)bya Eq. 535

The hourglass stiffness can be defined as1

bxixj

Nxi-------x

j

ij= =


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Chapter 5

Eq. 536

where

ais

anormalized to

||a||= 2, b = [b

x,b

y]T, and . Program Listing 53 implements the

hourglass element for heat conduction (project hourglass_heat in project workspace filefe.dsw).

We measure the time spent in the computation of stiffness matrix for the heat conduction element with irre-

ducible formulation in Chapter 4. It takes 3.5 seconds to assemble the stiffness on an obsolete 166 MHz PC. For

the mixed formulation in the last section, although we gain significant freedom in terms of formulation, the

assemble of diagonal and off-diagonal stiffness matrices (A and C) takes 6.2 seconds on the same computer. The

hourglass element takes only 0.5 second to assemble the stiffness.

1. see similar derivation for elasticity in p.251-254 in Hughes, T. J.R., The finite element method: linear static and dynamicfinite element analysis, Prentice-Hall, Inc., Englewood Cliffs, New Jersey, and references therein.

ke hourg la ss( )kJ b b( )

12----------------------- ( )

J det x

( )


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EP::element_pattern EP::ep = EP::QUADRILATERALS_4_NODES;

Omega_h::Omega_h() {

double coord[4][2] = {{0.0, 0.0}, {3.0, 0.0}, {3.0, 3.0}, {0.0, 3.0}};int control_node_flag[4] = {1, 1, 1, 1};

block(this, 4, 4, 4, control_node_flag, coord[0]);

}

gh_on_Gamma_h::gh_on_Gamma_h(int df, Omega_h& omega_h) {

__initialization(df, omega_h);

int row_node_no = 4;

for(int i = 0; i < row_node_no; i++) {

the_gh_array[node_order(i)](0) = gh_on_Gamma_h::Dirichlet;

the_gh_array[node_order(row_node_no*(row_node_no-1)+i)](0) =


the_gh_array[node_order(row_node_no*(row_node_no-1)+i)][0] = 30.0;}

}

class HourGlassHeatQ4 : public Element_Formulation { public:

HourGlassHeatQ4(Element_Type_Register a) : Element_Formulation(a) {}


HourGlassHeatQ4(int, Global_Discretization&);

};

Element_Formulation* HourGlassHeatQ4::make(int en, Global_Discretization& gd) {

return new HourGlassHeatQ4(en,gd);

}

HourGlassHeatQ4::HourGlassHeatQ4(int en, Global_Discretization& gd) :

Element_Formulation(en, gd) {


H1 Z(2, (double*)0, qp), Zai, Eta,

N = INTEGRABLE_VECTOR_OF_TANGENT_BUNDLE("int, int, Quadrature", 4, 2, qp);

Zai &= Z[0]; Eta &= Z[1];

N[0] = (1-Zai)*(1-Eta)/4; N[1] = (1+Zai)*(1-Eta)/4;

N[2] = (1+Zai)*(1+Eta)/4; N[3] = (1-Zai)*(1+Eta)/4;

H1 X = N*xl; H0 Nx = d(N) * d(X).inverse(); J dv(d(X).det());

double k_ = 1.0; C0 K_standard = (Nx * k_ * (~Nx)) | dv;

C0 h(4, (double*)0); h[3] = 1.0;C0 phi = h - Nx.quadrature_point_value(0)*((~xl)*h);

double factor = 2.0/norm(phi); phi *= factor;

H0 b = Nx(0) & Nx(1);

double j = (double)(d(X).det().quadrature_point_value(0));

C0 K_hourglass = (k_*j*((~b)*b) | dv) /12.0 * (phi%phi);

stiff &= K_standard + K_hourglass;

}


Element_Type_Register element_type_register_instance;

static HourGlassHeatQ4 hourglassheatq4_instance(element_type_register_instance);

int main() {

int ndf = 1; Omega_h oh; gh_on_Gamma_h gh(ndf, oh); U_h uh(ndf, oh);

Global_Discretization gd(oh, gh, uh); Matrix_Representation mr(gd);

mr.assembly();

C0 u = ((C0)(mr.rhs())) / ((C0)(mr.lhs()));

uh = u; uh = gh;

cout


18/172



Chapter 5

5.1.2 Mixed Formulation for Plane Elasticity

In the irreducible formulation for plane elasticity, the displacement field, u, is the only variable that the vari-

ation of the Lagrangian functional ( ), derived from the equilibrium equations, is taken as

Eq. 537

where the body force is denoted as b, the strain is defined as , and the differential operator L in matrix

form as

Eq. 538

The traction boundary condition is t = h on h. In the mixed formulation, in addition to the variational approxi-

mation on the equilibrium equations. we will also use the variational approximation to both the constitutiveequations and strain-displacement relations, separately. The interpolation functions (x) in finite elementapproximation of displacement field is taken as

Eq. 539

where is interpolation functions, and is nodal displacements. The subscript e denotes the element

level.

Hellinger-Reissner Variational Principle

In addition to the variational principle based on equilibrium equation in Eq. 537, we consider the constitu-

tive equation

Eq. 540

We also add stress field, , as additional variable in the Lagrangian functional such that

Eq. 541

Eq. 537 and Eq. 541 are the Euler-Lagrange equations corresponding to the Lagrangian functional.

Eq. 542

u( )

u( ) Lu( )T d

uT b d

uTh d

h

=

Lu=

L

x------ 0

0

y------

y------

x------

ue ea x( )uea

ea x( ) uea

D DLu= =

T Lu D 1 ( ) d 0=

u,( )1

2--- T

D

1

d uT

LT

b+( ) d uT

n h( ) dh+=


19/172



where t = n (the Cauchys formula). The Euler-Lagrange equations are obtained by taking the directional deriv-atives of the Lagrangian functional with respect to and u, then, make them equal to zero. Eq. 542 is known astheHellinger-Reissner variational principle. The finite element approximation of the stress field has interpola-

tion function

(x) in

Eq. 543

By inspecting Eq. 537 and Eq. 541, stress field has no derivatives taken on it. Therefore, the C0-continuity

requirement on the element boundaries, to ensure the integral equation does not give infinity, can be dropped.

The interpolation functions , in contrast to in Eq. 539, can be taken as piece-wise continuous func-

tions across the entire problem domain. For example, for four stress nodes taken at Gauss integration points with

the natural coordinates

Eq. 544

where a = 0, 1, 2, 3. The shape functions for these four nodes are

Eq. 545

If such discontinuous (at the element boundaries) interpolation is taken, the stress field can be approximated glo-

bally. Because there is no inter-element dependency. The subscript e on can be dropped. The matrix form ofEq. 537 and Eq. 541, at element level, is

Eq. 546

where

Eq. 547

Eq. 548

Eq. 549

Eq. 550

e ea x( )ea

ea x( ) ea x( )

a a,[ ]1

3-------

1

3-------,

1

3-------

1

3-------,

1

3-------

1

3-------,

1

3-------

1

3-------,, , ,

=

ea x( )1

4--- 1 3a+( ) 1 3a+( )

A CT

C 0

u

f1

f2

=

A D 1 ( ) de

=

C B de=

f1 A n( ) e CT

u eu=

f2 b d

e

n( ) de

C n( ) e+=

Ad d Fi i El M h dCh t 5


20/172



Chapter 5

where B = L. Again, if discontinuous interpolation on were taken, all terms involving dropped out in Eq.511 and Eq. 512, which are then significantly simplified. To avoid singular conditions as discussed in Eq. 518

we should have the condition to avoid singularity that

Eq. 551

Two quadrilateral elements with four -nodes and eight u-nodes (Q 4/8)1 are used to compute the beam bendingproblem in the higher-order path test in the Chapter 4. The numbers of degree of freedom for the two fields are

n = 8 3 = 24, nu = 13 2-4 = 22, which satisfied the conceptual patch test criterion in Eq. 551. The implemen-tation ofEq. 58 to Eq. 512 is shown in Program Listing 54 (project hellinger_reissner_variational_principle

in project workspace file fe.dsw). They present no new difficulty from the Program Listing 51. The solution

of the tip-deflection is 0.75, which is exact.

1. p.331 in Zienkiewicz, O.C. and R.L. Taylor, 1989, The finite element method, 4th ed., vol. 1. McGraw-Hill, Inc., UK.

n nu

Mi d d H b id Fi it El t M th d


21/172



1st element

coordinates of four corner nodes

physcial coordinates at ( )




2nd element

node # 8-20 are geometrical nodes

(serendipity)

1

3

-------1

3

-------,

1

3-------

1

3-------,

1

3-------

1

3-------,

1

3------- 1

3-------,


#include "include\omega_h_n.h"

Matrix_Representation_Couple::assembly_switch

Matrix_Representation_Couple::Assembly_Switch = Matrix_Representation_Couple::ALL;

static const int row_node_no = 5; static const int row_segment_no = row_node_no-1;

static const double L_ = 10.0; static const double c_ = 1.0;

static const double h_e_ = L_/((double)row_segment_no);

static const double E_ = 1.e3; static const double v_ = 0.3;

Omega_h_i::Omega_h_i( int i) : Omega_h(0) {

if(i == 0) {

double inv_sqrt3 = 1.0/sqrt(3.0), v[2], xl[4][2], zai, eta; Node *node;

xl[0][0] = 0.0; xl[0][1] = 0.0; xl[1][0] = 2.0*h_e_; xl[1][1] = 0.0;

xl[2][0] = 2.0*h_e_; xl[2][1] = 2.0*c_; xl[3][0] = 0.0; xl[3][1] = 2.0*c_;

zai = - inv_sqrt3; eta = - inv_sqrt3;

for(int j = 0; j < 2; j++) v[j] =(1.0-zai)*(1.0-eta)/4.0*xl[0][j]+(1.0+zai)*(1.0-eta)/4.0*xl[1][j]+(1.0+zai)*(1.0+eta)/4.0*xl[2][j]+ (1.0-zai)*(1.0+eta)/4.0*xl[3][j];

node = new Node(0, 2, v); node_array().add(node);

zai = inv_sqrt3; eta = - inv_sqrt3;

for(int j = 0; j < 2; j++) v[j] = (1.0-zai)*(1.0-eta)/4.0*xl[0][j]+ (1.0+zai)*(1.0-eta)/4.0*xl[1][j]+

(1.0+zai)*(1.0+eta)/4.0*xl[2][j]+ (1.0-zai)*(1.0+eta)/4.0*xl[3][j];


zai = inv_sqrt3; eta = inv_sqrt3;




zai = - inv_sqrt3; eta = inv_sqrt3;

for(int j = 0; j < 2; j++) v[j] = (1.0-zai)*(1.0-eta)/4.0*xl[0][j]+ 1.0+zai)*(1.0-eta)/4.0*xl[1][j]+



xl[0][0] = 2.0*h_e_; xl[0][1] = 0.0; xl[1][0] = 4.0*h_e_; xl[1][1] = 0.0;

xl[2][0] = 4.0*h_e_; xl[2][1] = 2.0*c_; xl[3][0] = 2.0*h_e_; xl[3][1] = 2.0*c_;





zai = inv_sqrt3; eta = - inv_sqrt3;for(int j = 0; j < 2; j++) v[j] = (1.0-zai)*(1.0-eta)/4.0*xl[0][j]+ (1.0+zai)*(1.0-eta)/4.0*xl[1][j]+











v[0] = 0.0; v[1] = 0.0; node = new Node(8, 2, v); node_array().add(node);

v[0] = 1.0*h_e_; node = new Node(9, 2, v); node_array().add(node);




v[0] = 0.0; v[1] = 1.0*c_; node = new Node(13, 2, v); node_array().add(node);





Advanced Finite Element MethodsChapter 5


22/172



Chapter 5

last four nodes are -nodes

B.C.disable all geometrical nodesu B.C.

fixed B.C. at right-end

Bending Moment on left-end




int ena[12]; Omega_eh *elem;

ena[0] = 8; ena[1] = 10; ena[2] = 18; ena[3] = 16; ena[4] = 9; ena[5] = 14;


elem = new Omega_eh(0, 0, 0, 12, ena); omega_eh_array().add(elem);




} else if(i == 1) { // Omega_u

double v[2]; Node *node;


v[0] = h_e_; node = new Node(1, 2, v); node_array().add(node);


v[0] = 3.0*h_e_; node = new Node(3, 2, v); node_array().add(node);v[0] = 4.0*h_e_; node = new Node(4, 2, v); node_array().add(node);










ena[0] = 0; ena[1] = 2; ena[2] = 10; ena[3] = 8; ena[4] = 1; ena[5] = 6; ena[6] = 9; ena[7] = 5;




}

}



if(i == 0) {

for(int j = 8; j


23/172



Q 4/8 element definition

diagonal A-matrix definition

Serendipity shape functions for integra-

ion and coordinate transformation

Step 1: initial four corner nodes

Step 2: add four edge nodes

Step 3: correction of four corner nodes

the presence of four edge nodes

with four nodes at four Gaussintegration points

N0-7 =0 (corresponding to geom. nodes)

N8-11 = a

N

D-1

off-diagonal C-matrix definition

A D 1 ( ) de

=

class ElasticQ84_Mixed_Formulation : public Element_Formulation_Couple {

public:

ElasticQ84_Mixed_Formulation(Element_Type_Register a):Element_Formulation_Couple(a) {}


ElasticQ84_Mixed_Formulation(int, Global_Discretization&);


ElasticQ84_Mixed_Formulation(int, Global_Discretization_Couple&);

};

Element_Formulation* ElasticQ84_Mixed_Formulation::make(int en,Global_Discretization& gd){

return new ElasticQ84_Mixed_Formulation(en,gd); }

static const double a_ = E_ / (1-pow(v_,2));

static const double Dv[3][3] = { {a_, a_*v_, 0.0 }, {a_*v_, a_, 0.0 }, {0.0, 0.0, a_*(1-v_)/2.0} };

C0 D = MATRIX("int, int, const double*", 3, 3, Dv[0]);

ElasticQ84_Mixed_Formulation::ElasticQ84_Mixed_Formulation(

int en, Global_Discretization& gd) : Element_Formulation_Couple(en, gd) {Quadrature qp(2, 9);

H1 Z(2, (double*)0, qp),

N = INTEGRABLE_VECTOR_OF_TANGENT_BUNDLE(int, int, Quadrature", 8, 2, qp),

Zai, Eta; Zai &= Z[0]; Eta &= Z[1];

N[0] = (1.0-Zai)*(1.0-Eta)/4.0; N[1] = (1.0+Zai)*(1.0-Eta)/4.0;

N[2] = (1.0+Zai)*(1.0+Eta)/4.0; N[3] = (1.0-Zai)*(1.0+Eta)/4.0;

N[4] = (1.0-Zai.pow(2))*(1.0-Eta)/2.0; N[5] = (1.0-Eta.pow(2))*(1.0+Zai)/2.0;

N[6] = (1.0-Zai.pow(2))*(1.0+Eta)/2.0; N[7] = (1.0-Eta.pow(2))*(1.0-Zai)/2.0;

N[0] = N[0] - (N[4]+N[7])/2.0; N[1] = N[1] - (N[4]+N[5])/2.0;

N[2] = N[2] - (N[5]+N[6])/2.0; N[3] = N[3] - (N[6]+N[7])/2.0;


H1 X = N*x; J dv(d(X).det());

double sqrt3 = sqrt(3.0); C0 zero(0.0);

H0 n = INTEGRABLE_VECTOR("int, Quadrature", 4/*nen*/, qp), n8, n9, n10, n11;

H0 zai, eta;

zai &= ((H0)Z[0]); eta &= ((H0)Z[1]);

n[0] = (1.0-sqrt3*zai)*(1.0-sqrt3*eta)/4.0; n[1] = (1.0+sqrt3*zai)*(1.0-sqrt3*eta)/4.0;

n[2] = (1.0+sqrt3*zai)*(1.0+sqrt3*eta)/4.0; n[3] = (1.0-sqrt3*zai)*(1.0+sqrt3*eta)/4.0;

n8 &= n[0]; n9 &= n[1]; n10 &= n[2]; n11 &= n[3];

H0 N_sig = ( (n8 | zero | zero | n9 | zero | zero | n10 | zero | zero | n11 | zero | zero ) &

(zero | n8 | zero | zero | n9 | zero | zero | n10 | zero | zero | n11 | zero ) &(zero |(zero | n8) | zero | zero | n9 | zero | zero | n10 | zero | zero | n11 ));

Matrix::Decomposition_Method = Matrix::Cholesky_Decomposition;

C0 D_inv = D.inverse();

Matrix::Decomposition_Method = Matrix::LU_Decomposition;

stiff &= MATRIX("int, int", 36, 36);

C0 stiff_sub = MATRIX("int, int, C0&, int, int", 12, 12, stiff, 24, 24);

stiff_sub = -((~N_sig) * (D_inv * N_sig)) | dv;

}

Element_Formulation_Couple* ElasticQ84_Mixed_Formulation::make(

int en, Global_Discretization_Couple& gdc) {return new ElasticQ84_Mixed_Formulation(en,gdc);

}


int en, Global_Discretization_Couple& gdc) : Element_Formulation_Couple(en, gdc) {


H1 Z(2, (double*)0, qp),

N = INTEGRABLE_VECTOR_OF_TANGENT_BUNDLE("int, int, Quadrature", 8, 2, qp),

Zai, Eta;

Zai &= Z[0]; Eta &= Z[1];

N[0] = (1.0-Zai)*(1.0-Eta)/4.0; N[1] = (1.0+Zai)*(1.0-Eta)/4.0;N[2] = (1.0+Zai)*(1.0+Eta)/4.0; N[3] = (1.0-Zai)*(1.0+Eta)/4.0;



24/172



Chapter 5



N[0] = N[0] - (N[4]+N[7])/2.0; N[1] = N[1] - (N[4]+N[5])/2.0;

N[2] = N[2] - (N[5]+N[6])/2.0; N[3] = N[3] - (N[6]+N[7])/2.0;

H1 X = N*xl;

H0 Nx = d(N) * d(X).inverse();

J dv(d(X).det());

H0 w_x = INTEGRABLE_SUBMATRIX("int, int, H0&", 1, 2, Nx),

wx, wy, B;

wx &= w_x[0][0]; wy &= w_x[0][1];

C0 zero(0.0);

B &= (~wx || zero) &

(zero || ~wy ) &

(~wy || ~wx );

double sqrt3 = sqrt(3.0);

H0 n = INTEGRABLE_VECTOR("int, Quadrature", 12/*nen*/, qp),n8, n9, n10, n11, zai, eta;

zai &= (H0)Z[0]; eta &= (H0)Z[1];



n8 &= n[0]; n9 &= n[1]; n10 &= n[2]; n11 &= n[3];

H0 N_sig = ((n8 | zero | zero | n9 | zero | zero | n10 | zero | zero | n11 | zero | zero ) &

(zero | n8 | zero | zero | n9 | zero | zero | n10 | zero | zero | n11 | zero ) &

(zero |(zero | n8) | zero | zero | n9 | zero | zero | n10 | zero | zero | n11 ));



stiff_sub = ((~B)*N_sig) | dv;

}


static Element_Type_Register element_type_register_instance;

static ElasticQ84_Mixed_Formulation

elasticq84_mixed_formulation_instance(element_type_register_instance);

static Matrix_Representation mr(sigma_gd);

static Matrix_Representation_Couple mrc(gdc, 0, 0, &(mr.rhs()), &mr);

int main() {

mr.assembly();

mrc.assembly();C0 A = ((C0)(mr.lhs())), f_1 = ((C0)(mr.rhs())),

C = ((C0)(mrc.lhs())), f_2 = ((C0)(mrc.rhs()));

Cholesky dnH(-A);

C0 Ainv = -(dnH.inverse());

C0 CAinvCt = C*Ainv*(~C);

LU dCAinvCt(CAinvCt);

cout


25/172



Hu-Washizu Variational Principle

In addition to Eq. 537 , the constitutive equations and the strain-displacement relations can be used such as

= D , and = Lu Eq. 552The variational approximation to these two equations are

Eq. 553

Eq. 553 and Eq. 537 are the Euler-Lagange equations of the Lagrangian functional

Eq. 554

The Lagrangian functional in Eq. 554 is known as the Hu-Washizu variational principle. The finite element

approximation to the strain field uses the interpolation functions (x) as

Eq. 555

The matrix form, at element level, ofEq. 537 and Eq. 553 is

Eq. 556

where

Eq. 557

Eq. 558

Eq. 559

Eq. 560

T D ( ) d 0= and T Lu ( ) d

0=,

, u,( ) 12--- TD d

T Lu( ) d

uTb d

uTh d

h

=

e ea x( )ea

A CT 0

C 0 ET

0 E 0

u

f1

f2

f3

=

A D( ) de

=

E B de=

C de

=

f1

A n

( )

eCT n

( )

e=



26/172


p

Eq. 561

Eq. 562

The condition for non-singular matrices is1

Eq. 563

The Program Listing 55 implements Eq. 556 to Eq. 562 (project hu_washizu_variational_principle inproject workspace file fe.dsw). With the same patch test problem for the Hellinger-Reissner variational princi-

ple in the previous section. We use shape functions with four nodes at Gaussian integration points for both stress

() and strain () fields; i.e., = . This results in C-matrix in Eq. 559 to be symmetrical negative definitive.Care should be taken, if Cholesky decomposition is used, which is applicable to a symmetricalpositive defini-

tive matrix. The displacement (u) shape function is an eight-nodes serendipity element. These choices satisfythe condition in Eq. 563. The coding of matrix substructuring technique supported by fe.lib becomes a little

more elaborated with three-fields (, , u) instead of two-fields (, u).The modification from two-fields problem is minor, however. For the definitions of discretized global

domain and boundary have three index entries as

1 Omega_h_i::Omega_h_i(int i) : Omega_h(0) { // hi2 if(i == 0 || i == 1) { // h or

h

3 ...

4 } else if(i == 2) { // hu

5 ...6 }

7 gh_on_Gamma_h_i::gh_on_Gamma_h_i(int i, int df, Omega_h& omega_h) : gh_on_Gamma_h() { //hi8 gh_on_Gamma_h::__initialization(df, omega_h);

9 if(i == 0) { // h10 ...

11 } else if(i == 1) { // h12 ...

13 } else if(i == 2) { // hu

14 ...

15 }

16 }

The instantiation of global discretized couples are

1. see discussion in p.333-334 in Zienkiewicz, O.C. and R.L. Taylor, 1989, The finite element method, 4th ed., vol. 1.McGraw-Hill, Inc., UK.

f2 C n( ) e ETu

eu

=

f3 b de n( ) d

e E n( ) e+=

n nu+ n and n nu,



27/172


define , and

elem # 0 nodal coordinates

1st Gauss point natural coordinates

1st Gauss point physical coordinates

2nd Gauss point natural coordinates

2nd Gauss point physical coordinates

3rd Gauss point natural coordinates3rd Gauss point physical coordinates

4th Gauss point natural coordinates

4th Gauss point physical coordinates

elem # 1 nodal coordinates1st Gauss point natural coordinates




3rd Gauss point natural coordinates

3rd Gauss point physical coordinates





Matrix_Representation_Couple::assembly_switch Matrix_Representation_Couple::

Assembly_Switch = Matrix_Representation_Couple::ALL;static const int row_node_no = 5; static const int row_segment_no = row_node_no-1;


static const double h_e_ = L_/((double)row_segment_no);


Omega_h_i::Omega_h_i(int i) : Omega_h(0) {

if(i == 0 || i == 1) {

double inv_sqrt3 = 1.0/sqrt(3.0), v[2], xl[4][2], zai, eta;

Node *node;

xl[0][0] = 0.0; xl[0][1] = 0.0; xl[1][0] = 2.0*h_e_; xl[1][1] = 0.0;

xl[2][0] = 2.0*h_e_; xl[2][1] = 2.0*c_; xl[3][0] = 0.0; xl[3][1] = 2.0*c_;


for(int j = 0; j < 2; j++)

v[j] = (1.0-zai)*(1.0-eta)/4.0*xl[0][j]+ (1.0+zai)*(1.0-eta)/4.0*xl[1][j]+




for(int j = 0; j < 2; j++)

v[j] = (1.0-zai)*(1.0-eta)/4.0*xl[0][j]+(1.0+zai)*(1.0-eta)/4.0*xl[1][j]+

(1.0+zai)*(1.0+eta)/4.0*xl[2][j]+(1.0-zai)*(1.0+eta)/4.0*xl[3][j];


zai = inv_sqrt3; eta = inv_sqrt3;for(int j = 0; j < 2; j++)





for(int j = 0; j < 2; j++)




xl[0][0] = 2.0*h_e_; xl[0][1] = 0.0; xl[1][0] = 4.0*h_e_; xl[1][1] = 0.0;

xl[2][0] = 4.0*h_e_; xl[2][1] = 2.0*c_; xl[3][0] = 2.0*h_e_; xl[3][1] = 2.0*c_;zai = - inv_sqrt3; eta = - inv_sqrt3;

for(int j = 0; j < 2; j++)





for(int j = 0; j < 2; j++)



node = new Node(5, 2, v); node_array().add(node);zai = inv_sqrt3; eta = inv_sqrt3;

for(int j = 0; j < 2; j++)





for(int j = 0; j < 2; j++)






28/172


geometrical nodes supply four corner

nodes coordinates to the element level

define elements

first 8 nodes are geometrical nodeslast four nodes are real nodes

u8-node serendipity element

define nodes

define elements

B.C.disable all geometrical nodes

B.C.disable all geometrical nodes

Wu boundary condition

Simpsons integration rule:

{-15 *2/3, 0*4/3, 15*2./3}

= {-5, 0, 5}
















ena[6] = 17; ena[7] = 13; ena[8] = 0; ena[9] = 1; ena[10] = 2; ena[11] = 3;elem = new Omega_eh(0, 0, 0, 12, ena); omega_eh_array().add(elem);




} else if(i == 2) {

double v[2]; Node *node;





v[0] = 4.0*h_e_; node = new Node(4, 2, v); node_array().add(node);v[0] = 0.0; v[1] = 1.0*c_; node = new Node(5, 2, v); node_array().add(node);













}

}



if(i == 0) {

for(int j = 8; j


29/172


g.d. for defining A matrixh = {x, y, }T

h = {x, y, }T

u , uh = {u, v}T

g.d. u for defining irreducible K matrix

g.d.couple {-} for defining C matrix

g.d.couple {u-} for defining E matrix

diagonals A & K matrices

off-diagonals C & E matrices

diagonals matrices

A matrix formulation

eight nodes serendipity shape function

h = {x, y, }T

K matrix; for iterative method to esti-

mate initial u values

A D( ) de

=

static const int epsilon_ndf = 3; static Omega_h_i oh_epsilon(0);

static gh_on_Gamma_h_i epsilon_gh(0, epsilon_ndf, oh_epsilon);

static U_h epsilon_h(epsilon_ndf, oh_epsilon);

static Global_Discretization *epsilon_type = new Global_Discretization;

static Global_Discretization epsilon_gd(oh_epsilon, epsilon_gh, epsilon_h, epsilon_type);

static const int sigma_ndf = 3; static Omega_h_i oh_sigma(1);static gh_on_Gamma_h_i sigma_gh(1, sigma_ndf, oh_sigma);

static U_h sigma_h(sigma_ndf, oh_sigma);

static Global_Discretization sigma_gd(oh_sigma, sigma_gh, sigma_h);

static const int u_ndf = 2; static Omega_h_i oh_u(2);

static gh_on_Gamma_h_i u_gh(2, u_ndf, oh_u); static U_h u_h(u_ndf, oh_u);

static Global_Discretization *u_type = new Global_Discretization;

static Global_Discretization u_gd(oh_u, u_gh, u_h, u_type);

static Global_Discretization_Couple *sigma_epsilon = new Global_Discretization_Couple();

static Global_Discretization_Couple gdc_sigma_epsilon(sigma_gd, epsilon_gd, sigma_epsilon);

static Global_Discretization_Couple *u_sigma = new Global_Discretization_Couple();

static Global_Discretization_Couple gdc_u_sigma(u_gd, sigma_gd, u_sigma);

class ElasticQ84_Mixed_Formulation : public Element_Formulation_Couple {

public:

ElasticQ84_Mixed_Formulation(Element_Type_Register a):Element_Formulation_Couple(a) {}


ElasticQ84_Mixed_Formulation(int, Global_Discretization&);


ElasticQ84_Mixed_Formulation(int, Global_Discretization_Couple&);

};

Element_Formulation* ElasticQ84_Mixed_Formulation::make(

int en,Global_Discretization& gd) { return new ElasticQ84_Mixed_Formulation(en,gd); }static const double a_ = E_ / (1-pow(v_,2));

static const double Dv[3][3] = { {a_, a_*v_, 0.0 }, {a_*v_, a_, 0.0}, {0.0, 0.0, a_*(1-v_)/2.0} };

C0 D = MATRIX("int, int, const double*", 3, 3, Dv[0]);


int en, Global_Discretization& gd) : Element_Formulation_Couple(en, gd) {

if(gd.type() == epsilon_type) {




Zai &= Z[0]; Eta &= Z[1];

N[0] = (1.0-Zai)*(1.0-Eta)/4.0; N[1] = (1.0+Zai)*(1.0-Eta)/4.0;

N[2] = (1.0+Zai)*(1.0+Eta)/4.0; N[3] = (1.0-Zai)*(1.0+Eta)/4.0;



N[0] = N[0] - (N[4]+N[7])/2.0; N[1] = N[1] - (N[4]+N[5])/2.0;

N[2] = N[2] - (N[5]+N[6])/2.0; N[3] = N[3] - (N[6]+N[7])/2.0;


H1 X = N*x; J dv(d(X).det()); double sqrt3 = sqrt(3.0); C0 zero(0.0);

H0 n = INTEGRABLE_VECTOR("int, Quadrature", 4, qp), n8, n9, n10, n11, zai, eta;

zai &= ((H0)Z[0]); eta &= ((H0)Z[1]);

n[0] = (1.0-sqrt3*zai)*(1.0-sqrt3*eta)/4.0; n[1] = (1.0+sqrt3*zai)*(1.0-sqrt3*eta)/4.0;n[2] = (1.0+sqrt3*zai)*(1.0+sqrt3*eta)/4.0; n[3] = (1.0-sqrt3*zai)*(1.0+sqrt3*eta)/4.0;

n8 &= n[0]; n9 &= n[1]; n10 &= n[2]; n11 &= n[3];

H0 N_epsilon = ((n8 | zero | zero | n9 | zero | zero | n10 | zero | zero | n11 | zero | zero ) &





stiff_sub = ((~N_epsilon) * (D * N_epsilon)) | dv;

} else {


H1 Z(2, (double*)0, qp), , Zai, Eta

N = INTEGRABLE_VECTOR_OF_TANGENT_BUNDLE( "int, int, Quadrature", 8, 2, qp);



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K is a standard stiffness matrix

off-diagonal matrices

C matrux formulation

h

= {x, y, }T

E matrux formulation

C de

=

Zai &= Z[0]; Eta &= Z[1];

N[0] = (1.0-Zai)*(1.0-Eta)/4.0; N[1] = (1.0+Zai)*(1.0-Eta)/4.0;

N[2] = (1.0+Zai)*(1.0+Eta)/4.0; N[3] = (1.0-Zai)*(1.0+Eta)/4.0;



N[0] -= (N[4]+N[7])/2.0; N[1] -= (N[4]+N[5])/2.0;N[2] -= (N[5]+N[6])/2.0; N[3] -= (N[6]+N[7])/2.0;


H0 w_x = INTEGRABLE_SUBMATRIX("int, int, H0&", 1, 2, Nx), wx, wy, B;

wx &= w_x[0][0]; wy &= w_x[0][1]; C0 zero(0.0);

B &= (~wx || zero) & (zero || ~wy ) & (~wy || ~wx );

stiff &= ((~B)*D*B)|dv;

}

}

Element_Formulation_Couple* ElasticQ84_Mixed_Formulation::make(int en,

Global_Discretization_Couple& gdc) { return new ElasticQ84_Mixed_Formulation(en,gdc); }ElasticQ84_Mixed_Formulation::ElasticQ84_Mixed_Formulation(int en,

Global_Discretization_Couple& gdc) : Element_Formulation_Couple(en, gdc) {

if(gdc.type() == sigma_epsilon) {



N = INTEGRABLE_VECTOR_OF_TANGENT_BUNDLE("int, int, Quadrature",8,2, qp);

Zai &= Z[0]; Eta &= Z[1];

N[0] = (1.0-Zai)*(1.0-Eta)/4.0; N[1] = (1.0+Zai)*(1.0-Eta)/4.0;

N[2] = (1.0+Zai)*(1.0+Eta)/4.0; N[3] = (1.0-Zai)*(1.0+Eta)/4.0;


N[6] = (1.0-Zai.pow(2))*(1.0+Eta)/2.0; N[7] = (1.0-Eta.pow(2))*(1.0-Zai)/2.0;N[0] = N[0] - (N[4]+N[7])/2.0; N[1] = N[1] - (N[4]+N[5])/2.0;

N[2] = N[2] - (N[5]+N[6])/2.0; N[3] = N[3] - (N[6]+N[7])/2.0;

H1 X = N*xl; J dv(d(X).det());

double sqrt3 = sqrt(3.0); C0 zero(0.0);


zai &= ((H0)Z[0]); eta &= ((H0)Z[1]);



n8 &= n[0]; n9 &= n[1]; n10 &= n[2]; n11 &= n[3];

H0 N_sig = ((n8 | zero | zero | n9 | zero | zero | n10 | zero | zero | n11 | zero | zero ) &



H0 N_epsilon; N_epsilon &= N_sig;



stiff_sub = -((~N_sig)*N_epsilon) | dv;

} else {



N = INTEGRABLE_VECTOR_OF_TANGENT_BUNDLE( "int, int, Quadrature",8,2, qp);

Zai &= Z[0]; Eta &= Z[1];N[0] = (1.0-Zai)*(1.0-Eta)/4.0; N[1] = (1.0+Zai)*(1.0-Eta)/4.0;

N[2] = (1.0+Zai)*(1.0+Eta)/4.0; N[3] = (1.0-Zai)*(1.0+Eta)/4.0;



N[0] = N[0] - (N[4]+N[7])/2.0; N[1] = N[1] - (N[4]+N[5])/2.0;

N[2] = N[2] - (N[5]+N[6])/2.0; N[3] = N[3] - (N[6]+N[7])/2.0;


H0 w_x = INTEGRABLE_SUBMATRIX("int, int, H0&", 1, 2, Nx), wx, wy, B;

wx &= w_x[0][0]; wy &= w_x[0][1]; C0 zero(0.0);

B &= (~wx || zero) &

(zero || ~wy ) &

(~wy || ~wx );



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zai &= ((H0)Z[0]); eta &= ((H0)Z[1]);



n8 &= n[0]; n9 &= n[1]; n10 &= n[2]; n11 &= n[3];H0 N_sig = ((n8 | zero | zero | n9 | zero | zero | n10 | zero | zero | n11 | zero | zero ) &





stiff_sub = ((~B)*N_sig) | dv;

}

}


static Element_Type_Register element_type_register_instance;

static ElasticQ84_Mixed_Formulation

elasticq84_mixed_formulation_instance(element_type_register_instance);

static Matrix_Representation mr(epsilon_gd);

static Matrix_Representation_Couple mrcC(gdc_sigma_epsilon, 0, 0, &(mr.rhs()), &mr);

static Matrix_Representation_Couple mrcE(gdc_u_sigma, 0, 0, &(mrcC.rhs()), &mrcC);

int main() {

mrcC.assembly();

mr.assembly();

mrcE.assembly();

C0 A = ((C0)(mr.lhs())),

f_1 = ((C0)(mr.rhs())),C = ((C0)(mrcC.lhs())),

f_2 = ((C0)(mrcC.rhs())),

E = ((C0)(mrcE.lhs())),

f_3 = ((C0)(mrcE.rhs()));

Cholesky dnC(-C);

C0 Cinv = -(dnC.inverse()),

CinvA = Cinv*A,

CinvACinv = Cinv*A*Cinv,

ECinvACinv = E*CinvACinv,

ECinvACinvEt = ECinvACinv*(~E);

Cholesky dECinvACinvEt(ECinvACinvEt);

C0 u = dECinvACinvEt * (f_3 - (E*Cinv)*f_1 + ECinvACinv*f_2);

C0 epsilon = -(dnC*(f_2-(~E)*u));

C0 sigma = -(dnC*(f_1-A*epsilon));

epsilon_h = epsilon;

epsilon_h = epsilon_gd.gh_on_gamma_h();

cout


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1 static const int epsilon_ndf = 3; // ndf 2 static Omega_h_i oh_epsilon(0); // h3 static gh_on_Gamma_h_i epsilon_gh(0, epsilon_ndf, oh_epsilon); // h4 static U_h epsilon_h(epsilon_ndf, oh_epsilon); // x, yxy5 static Global_Discretization *epsilon_type = new Global_Discretization;

6 static Global_Discretization epsilon_gd(oh_epsilon, epsilon_gh, epsilon_h, epsilon_type);

7 static const int sigma_ndf = 3; // ndf 8 static Omega_h_i oh_sigma(1); // h9 static gh_on_Gamma_h_i sigma_gh(1, sigma_ndf, oh_sigma); // h10 static U_h sigma_h(sigma_ndf, oh_sigma); // x, yxy11 static Global_Discretization sigma_gd(oh_sigma, sigma_gh, sigma_h);

12 static const int u_ndf = 2; // ndf u

13 static Omega_h_i oh_u(2); // hu14 static gh_on_Gamma_h_i u_gh(2, u_ndf, oh_u); // hu15 static U_h u_h(u_ndf, oh_u); // ux, uy16 static Global_Discretization *u_type = new Global_Discretization;

17 static Global_Discretization u_gd(oh_u, u_gh, u_h, u_type);

18 static Global_Discretization_Couple *sigma_epsilon = new Global_Discretization_Couple(); //19 static Global_Discretization_Couple gdc_sigma_epsilon(sigma_gd, epsilon_gd, sigma_epsilon);

20 static Global_Discretization_Couple *u_sigma = new Global_Discretization_Couple(); // u-

21 static Global_Discretization_Couple gdc_u_sigma(u_gd, sigma_gd, u_sigma);

Three pointers to Global_Discretization and Global_Discretization_Coupe, epsilon_type, u_type, and

sigma_epsilon, are to be used in Element_Formulation class definitions to switch to corresponding segments of

submatrices definitions.

1 class ElasticQ84_Mixed_Formulation : public Element_Formulation_Couple {

2 public:3 ElasticQ84_Mixed_Formulation(Element_Type_Register a) : Element_Formulation_Couple(a) {}

4 Element_Formulation *make(int, Global_Discretization&); // diagonal submatrices

5 ElasticQ84_Mixed_Formulation(int, Global_Discretization&);

6 Element_Formulation_Couple *make(int, Global_Discretization_Couple&);// off-diagonal submatrices

7 ElasticQ84_Mixed_Formulation(int, Global_Discretization_Couple&);

8 };

9 Element_Formulation* ElasticQ84_Mixed_Formulation::make(int en, Global_Discretization& gd) {

10 return new ElasticQ84_Mixed_Formulation(en,gd);11 }

12 ElasticQ84_Mixed_Formulation::ElasticQ84_Mixed_Formulation(int en, Global_Discretization& gd) :

13 Element_Formulation_Couple(en, gd) {

14 if(gd.type() == epsilon_type) { // define A-Matrix; field15 ...

16 } else { //K-matrix; u-field, for iterative method to estimate initial u values, Z&T p. 361, Eq. 12.99

17 ...

18 }19 }



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20 Element_Formulation_Couple* ElasticQ84_Mixed_Formulation::make(

21 int en, Global_Discretization_Couple& gdc) { return new ElasticQ84_Mixed_Formulation(en,gdc); }

22 ElasticQ84_Mixed_Formulation::ElasticQ84_Mixed_Formulation(

23 int en, Global_Discretization_Couple& gdc) : Element_Formulation_Couple(en, gdc) {

24 if(gdc.type() == sigma_epsilon) { // define C-Matrix; fields25 ...

26 } else { // define E-matrix; u- fields27 ...

28 }

29 }

Line 16 is to construct the convergence acceleration matrix if iterative method is used.1 The global solution is

proceeded with to be solved first as

= (EC-1 AC-1 ET)-1 (f3 - EC-1 f1 + EC

-1 AC-1 f2) Eq. 564

After is obtained, we can substituting in the following equation

= C-1 (f2 - ET ) Eq. 565

Then, is solved by substituting in

= C-1 (f1 - A ) Eq. 566

The Matrix_Representations are in instantiated in the followings with the global substructuring solution

1 int main() {

2 Matrix_Representation mr(epsilon_gd);

3 Matrix_Representation_Couple mrcC(gdc_sigma_epsilon, 0, 0, &(mr.rhs()), &mr);

4 Matrix_Representation_Couple mrcE(gdc_u_sigma, 0, 0, &(mrcC.rhs()), &mrcC);

5 mrcC.assembly();

6 mr.assembly();

7 mrcE.assembly();

8 C0 A = ((C0)(mr.lhs())), f_1 = ((C0)(mr.rhs())),

9 C = ((C0)(mrcC.lhs())), f_2 = ((C0)(mrcC.rhs())),

10 E = ((C0)(mrcE.lhs())), f_3 = ((C0)(mrcE.rhs()));

11 Cholesky dnC(-C); // decomposition; C is symmetrical negative definite

12 C0 Cinv = -(dnC.inverse()); // = (EC-1AC-1ET)-1 (f3-EC-1f1+ EC

-1AC-1f2)

13 C0 CinvA = Cinv*A;

14 C0 CinvACinv = CinvA*Cinv;

1. see next section and p. 361 in Zienkiewicz, O.C. and R.L. Taylor, 1989, The finite element method, 4th ed., vol. 1.McGraw-Hill, Inc., UK.

u

u

u u

u

u



34/172


15 C0 ECinvACinv = E*CinvACinv;

16 C0 ECinvACinvEt = ECinvACinv*(~E);

17 Cholesky dECinvACinvEt(ECinvACinvEt);

18 C0 u = dECinvACinvEt * (f_3 - (E*Cinv)*f_1 + ECinvACinv*f_2);

19 C0 epsilon = -(dnC*(f_2-(~E)*u)); // = C-1(f2-ET )

20 C0 sigma = -(dnC*(f_1-A*epsilon)); // = C-1(f1-A )

21 epsilon_h = epsilon; // update free degree of freedom

22 epsilon_h = epsilon_gd.gh_on_gamma_h(); // update fixed degree of freedom

23 cout


35/172



36/172


, and Eq. 576

Substituting into Eq. 575 and Eq. 572 gives in matrix form at element level is

Eq. 577

where

Eq. 578

Eq. 579

Eq. 580

Eq. 581

Eq. 582

The solution to substructuring ofEq. 577 is proceeded from its second equation, using symmetrical negative

definitiveness ofV,

Eq. 583

Substituting into the first equation ofEq. 577 gives

Eq. 584

That is,

Eq. 585

Therefore, we first solve , and then substituting back to Eq. 583 for recovering . Program Listing 56

(project incompressible_u_p_formulation in project workspace file fe.dsw) implements the Q 4/9 element

(with an additional center u-node) and test problem in the higher-order patch test.

u Nuu= p Npp=

A CT

C V

u

p

f1

f2

=

A BT D02

3---m m Bd

e

=

C mNp( )TBd

e

=

VNp Np

K---------------------d

e

=

f1 NuTb d

e

NuTt de

Au e CTp e

+=

f2 Cu eVp e

=

p V 1 f2 Cu( )=

Au CTV 1 f2 Cu( )+ f1=

u A CTV 1 C[ ] 1 f1 CTV 1 f2[ ]=

u u p



37/172


plane stress modification

define u1st row

2nd row

3rd row

Lagrangian 9-nodes element

define pelem # 0 nodal coordinates











Matrix_Representation_Couple::assembly_switch

Matrix_Representation_Couple::Assembly_Switch = Matrix_Representation_Couple::ALL;

static const int row_node_no = 5; static const int row_segment_no = row_node_no-1;


static const double h_e_ = L_/((double)row_segment_no); // half element size


static const double lambda_ = v_*E_/((1+v_)*(1-2*v_));

static const double mu_ = E_/(2*(1+v_));

static const double lambda_bar = 2*lambda_*mu_/(lambda_+2*mu_);

static const double K_ = lambda_bar+2.0/3.0*mu_;

Omega_h_i::Omega_h_i(int i) : Omega_h(0) {

if(i == 0) { double v[2]; Node *node;


v[0] = h_e_; node = new Node(1, 2, v); node_array().add(node);v[0] = 2.0*h_e_; node = new Node(2, 2, v); node_array().add(node);














ena[0] = 0; ena[1] = 2; ena[2] = 12; ena[3] = 10; ena[4] = 1;

ena[5] = 7; ena[6] = 11; ena[7] = 5; ena[8] = 6;


ena[0] = 2; ena[1] = 4; ena[2] = 14; ena[3] = 12; ena[4] = 3;

ena[5] = 9; ena[6] = 13; ena[7] = 7; ena[8] = 8;


} else if(i == 1) { double inv_sqrt3 = 1.0/sqrt(3.0), v[2], xl[4][2], zai, eta; Node *node;xl[0][0] = 0.0; xl[0][1] = 0.0; xl[1][0] = 2.0*h_e_; xl[1][1] = 0.0;

xl[2][0] = 2.0*h_e_; xl[2][1] = 2.0*c_; xl[3][0] = 0.0; xl[3][1] = 2.0*c_;


for(int j = 0; j < 2; j++)





for(int j = 0; j < 2; j++)





for(int j = 0; j < 2; j++)





for(int j = 0; j < 2; j++)


(1.0+zai)*(1.0+eta)/4.0*xl[2][j]+ (1.0-zai)*(1.0+eta)/4.0*xl[3][j];node = new Node(3, 2, v); node_array().add(node);


elem # 1 nodal coordinatesxl[0][0] = 2 0*h e ; xl[0][1] = 0 0; xl[1][0] = 4 0*h e ; xl[1][1] = 0 0;


38/172


elem # 1 nodal coordinates









geometrical nodes

last four nodes are real pressure-field

nodes

Dirichlet B.C.

Nuemann B.C.

disable all geometrical nodes

xl[0][0] = 2.0 h_e_; xl[0][1] = 0.0; xl[1][0] = 4.0 h_e_; xl[1][1] = 0.0;

xl[2][0] = 4.0*h_e_; xl[2][1] = 2.0*c_; xl[3][0] = 2.0*h_e_; xl[3][1] = 2.0*c_;


for(int j = 0; j < 2; j++)


(1.0+zai)*(1.0+eta)/4.0*xl[2][j]+ (1.0-zai)*(1.0+eta)/4.0*xl[3][j];node = new Node(4, 2, v); node_array().add(node);


for(int j = 0; j < 2; j++)





for(int j = 0; j < 2; j++)





for(int j = 0; j < 2; j++)








v[0] = 4.0*h_e_; node = new Node(12, 2, v); node_array().add(node);v[0] = 0.0; v[1] = 1.0*c_; node = new Node(13, 2, v); node_array().add(node);












ena[6] = 19; ena[7] = 13; ena[8] = 14; ena[9] = 0; ena[10] = 1; ena[11] = 2; ena[12] = 3;



ena[6] = 21; ena[7] = 15; ena[8] = 16; ena[9] = 4; ena[10] = 5; ena[11] = 6; ena[12] = 7;


}

}


gh_on_Gamma_h::__initialization(df, omega_h);if(i == 0) {

the_gh_array[node_order(4)](0) = the_gh_array[node_order(9)](0) =

the_gh_array[node_order(14)](0)=the_gh_array[node_order(9)](1)=


the_gh_array[node_order(10)][0] = -5.0; the_gh_array[node_order(0)][0] = 5.0;

} else if(i == 1) {

for(int j = 8; j


39/172


u

u = [u, v]T

p

{p-u} global discretization couple

diagonal submatrices

Lagragian 9-node element

A BT D02

3---m m Bd

e

=

VNp Np

K---------------------d

e

=

static gh_on_Gamma_h_i u_gh(0, u_ndf, oh_u); static U_h u_h(u_ndf, oh_u);

static const int p_ndf = 1; static Omega_h_i oh_p(1);

static gh_on_Gamma_h_i p_gh(1, p_ndf, oh_p); static U_h p_h(p_ndf, oh_p);

static Global_Discretization *u_type = new Global_Discretization;

static Global_Discretization u_gd(oh_u, u_gh, u_h, u_type);static Global_Discretization *p_type = new Global_Discretization;

static Global_Discretization p_gd(oh_p, p_gh, p_h, p_type);

static Global_Discretization_Couple *p_u_type = new Global_Discretization_Couple;

static Global_Discretization_Couple gdc(p_gd, u_gd, p_u_type);

class Incompressible_ElasticQ94_Mixed_Formulation : public Element_Formulation_Couple {

public:

Incompressible_ElasticQ94_Mixed_Formulation(Element_Type_Register a) :

Element_Formulation_Couple(a) {}


Incompressible_ElasticQ94_Mixed_Formulation(int, Global_Discretization&);


Incompressible_ElasticQ94_Mixed_Formulation(int, Global_Discretization_Couple&);

};

Element_Formulation*

Incompressible_ElasticQ94_Mixed_Formulation::make(int en, Global_Discretization& gd) {

return new Incompressible_ElasticQ94_Mixed_Formulation(en,gd);

}

Incompressible_ElasticQ94_Mixed_Formulation::Incompressible_ElasticQ94_Mixed_Formulation

(int en, Global_Discretization& gd) : Element_Formulation_Couple(en, gd) {


H1 Z(2, (double*)0, qp), Zai, Eta,N = INTEGRABLE_VECTOR_OF_TANGENT_BUNDLE( "int, int, Quadrature", 9, 2, qp);

Zai &= Z[0]; Eta &= Z[1];

N[0] = (1-Zai)*(1-Eta)/4; N[1] = (1+Zai)*(1-Eta)/4;

N[2] = (1+Zai)*(1+Eta)/4; N[3] = (1-Zai)*(1+Eta)/4;

N[8] = (1-Zai.pow(2))*(1-Eta.pow(2));

N[0] -= N[8]/4; N[1] -= N[8]/4; N[2] -= N[8]/4; N[3] -= N[8]/4;

N[4] = ((1-Zai.pow(2))*(1-Eta)-N[8])/2; N[5] = ((1-Eta.pow(2))*(1+Zai)-N[8])/2;

N[6] = ((1-Zai.pow(2))*(1+Eta)-N[8])/2; N[7] = ((1-Eta.pow(2))*(1-Zai)-N[8])/2;

N[0] -= (N[4]+N[7])/2; N[1] -= (N[4]+N[5])/2; N[2] -= (N[5]+N[6])/2; N[3] -= (N[6]+N[7])/2;


H1 X = N*x; J dv(d(X).det());

if(gd.type() == u_type) {

H0 Nx = d(N) * d(X).inverse();

double d_0[3][3] = { {2.0, 0.0, 0.0}, {0.0, 2.0, 0.0}, {0.0, 0.0, 1.0}};

C0 D_0 = MATRIX("int, int, const double*", 3, 3, d_0[0]);

double m_0[3] = {1.0, 1.0, 0.0}; C0 m = VECTOR("int, const double*", 3, m_0);

H0 W_x = INTEGRABLE_SUBMATRIX("int, int, H0&", 1, nsd, Nx), Wx, Wy, B;

Wx &= W_x[0][0]; Wy &= W_x[0][1];

B &= (~Wx || C0(0.0) ) &

(C0(0.0) || ~Wy ) &

(~Wy || ~Wx );stiff &= ((~B) * (mu_*(D_0-2.0/3.0*(m%m)) * B)) | dv;

} else if(gd.type() == p_type) {


H0 n = INTEGRABLE_VECTOR("int, Quadrature", 4/*nen*/, qp), zai, eta;

zai &= ((H0)Z[0]); eta &= ((H0)Z[1]);





stiff_sub = -((n * (~n)) | dv)/K_;

}}


Element_Formulation_Couple* Incompressible_ElasticQ94_Mixed_Formulation::make(


40/172


int en, Global_Discretization_Couple& gdc) {

return new Incompressible_ElasticQ94_Mixed_Formulation(en,gdc);

}

Incompressible_ElasticQ94_Mixed_Formulation::Incompressible_ElasticQ94_Mixed_Formulation

(int en, Global_Discretization_Couple& gdc) : Element_Formulation_Couple(en, gdc) {Quadrature qp(2, 9);



Zai &= Z[0]; Eta &= Z[1];

N[0] = (1-Zai)*(1-Eta)/4; N[1] = (

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