*** DESCRIPTIONS OF EXAMPLE 202 *** Numerically Integrated Anisotropic Poisson Equation in 2-D or Axisymmetric K_xx U,xx + 2K_xy U,xy + K_yy U,yy + Q = 0 PROP(1) = CONDUCTIVITY K_XX PROP(2) = CONDUCTIVITY K_YY PROP(3) = CONDUCTIVITY K_XY PROP(4) = SOURCE PER UNIT VOLUME, Q or use my_exact_source_inc when EXACT_CASE = 0 DATA_SET = 01 EXACT_CASE = 02 Lakhany & Whiteman 2nd & 3rd deriv test, 200 T6 elements ! Solution of the 2-D test case used by Lakhany & Whiteman ! (u,xx+u,yy) = -(2 + pi_sq * (1-x^2)) * Sin (pi y) ! on (-1,-1)X(1,1). Exact u(x, y) = (1 - x^2) * Sin (pi y) ! For 2nd, 3rd derivative recovery via SCP recovery. ! The solution is symmetric about y = 0, and x = 0 ! "Superconvergeny Derivative Operators: Derivative Recovery ! Techniques", A. M. Lakhany, J. R. Whiteman, pp. 196-216, ! Finite Element Methods, M. Krizek, et. al (Eds), Marcel ! Decker, New York, 1998 ! The source term is complicated. C sensitive to gauss input ! gauss MAXIMUM, NODE MINIMUM, NODE ! 4 9.9975E-01, 326 -9.9975E-01, 116 ! 6 ! exact 1.0000000 326 -1.0000000 116 ! K_xx U,xx + 2K_xy U,xy + K_yy U,yy + Q = 0 ! Properties: K_xx, K_yy, K_xy, and if el_real >3, Q ! Here el_real = 3 so Q is obtained from the ! LAKHANY_WHITEMAN_TEST_SOURCE analytic equation ! which requires a higher order quadrature rule ! for the column matrix DATA_SET = 02 EXACT_CASE = 05 72 Q9 elements. Here essential bc is given on all edges. ! Solution of the 2-D Poisson Equation on an segment of a circular ! annulus in the first quadrant from r=1 to r=2 ! r = 2 |-----\ ! q_n = 0..| \ .... q_n = 4 X^2 Y^2 ! r = 1 |--\ K,Q \ Q = 4(X^2 + y^2) = 4 R^2 ! T=10-X^2+X^4...\ | K = 2 ! Y | | T(X,Y) = 10 - X^2 Y^2, exact ! o-X ------- ... T = 10 (10 is arbitrary constant) ! Dirchlet BC: constant at Y=0, variable at R=1 ! Normal flux: variable on R = 2, Zero on X = 0 (or on X = Y) ! ! Exact bc values will over-ride null input values ! PROP(1) = CONDUCTIVITY K_XX 1_D, 2_D, & 3_D ! PROP(2) = CONDUCTIVITY K_YY 2_D, & 3_D ! PROP(3) = CONDUCTIVITY K_XY, OR K_ZZ IF 3_D DATA_SET = 03 EXACT_CASE = 05 72 Q9 elements. Here exact flux used on R = 2, NBC at X = 0, EBC elsewhere ! Solution of the 2-D Poisson Equation on an segment of a circular ! annulus in the first quadrant from r=1 to r=2 ! r = 2 |-----\ ! q_n = 0..| \ .... q_n = 4 K X^2 Y^2 / R ! r = 1 |--\ K,Q \ Q = 4(X^2 + y^2) = 4 R^2 ! T=10-X^2+X^4...\ | K = 2 ! Y | | T(X,Y) = 10 - X^2 Y^2, exact ! o-X ------- ... T = 10 (10 is arbitrary constant) ! Dirchlet BC: constant at Y=0, variable at R=1 ! Normal flux: variable on R = 2, Zero on X = 0 (or on X = Y) ! Exact bc values will over-ride null input values below ! Note: on a constant radius R the normal flux is ! q_n = - 4.d0 * K * (X**2 * Y**2) / R ! For constant A angle line the normal flux is ! q_n = - 2.d0 * K * X * Y * (X * Sin A - Y * Cos A) ! PROP(1) = CONDUCTIVITY K_XX 1_D, 2_D, & 3_D ! PROP(2) = CONDUCTIVITY K_YY 2_D, & 3_D ! PROP(3) = CONDUCTIVITY K_XY, OR K_ZZ IF 3_D ! Normal flux exact values substituted on input. ! Version to substitute at QP is data_set 9 DATA_SET 04 EXACT_CASE = 05 72 Q9 elements. ??? same as 03 ??? Here exact flux used on R = 2, NBC at X = 0, EBC elsewhere ! Solution of the 2-D Poisson Equation on an segment of a circular ! annulus in the first quadrant from r=1 to r=2 ! r = 2 |-----\ ! q_n = 0..| \ .... q_n = 4 K X^2 Y^2 / R ! r = 1 |--\ K,Q \ Q = 4(X^2 + y^2) = 4 R^2 ! T=10-X^2+X^4...\ | K = 2 ! Y | | T(X,Y) = 10 - X^2 Y^2, exact ! o-X ------- ... T = 10 (10 is arbitrary constant) ! Dirchlet BC: constant at Y=0, variable at R=1 ! Normal flux: variable on R = 2, Zero on X = 0 (or on X = Y) ! Exact bc values will over-ride null input values below ! Note: on a constant radius R the normal flux is ! q_n = - 4.d0 * K * (X**2 * Y**2) / R ! For constant A angle line the normal flux is ! q_n = - 2.d0 * K * X * Y * (X * Sin A - Y * Cos A) ! PROP(1) = CONDUCTIVITY K_XX 1_D, 2_D, & 3_D ! PROP(2) = CONDUCTIVITY K_YY 2_D, & 3_D ! PROP(3) = CONDUCTIVITY K_XY, OR K_ZZ IF 3_D DATA_SET 05 EXACT_CASE = 08 Patch Test with Constant 2nd Derivs 96 T6 elements DATA_SET 06 EXACT_CASE = 0 Patch Test with Constant 1st Derivs 4 Q8 elements DATA_SET 07 EXACT_CASE = 0 (none) Segerlind Offset cylinder test of convection 221 T6 triangles, 27 L3 convection DATA_SET 08 EXACT_CASE = 20 ! Cubic 2d Laplace equation. Q=0, Kx=Ky=1, Kxy=0 ! u,xx + u,yy = 0, u=-x^3 -y^3 + 3x^2y + 3xy^2 ! K_xx U,xx + 2K_xy U,xy + K_yy U,yy + Q = 0 ! Properties: K_xx, K_yy, K_xy, and if el_real >3, Q ! 96 T6 elements DATA_SET = 09 EXACT_CASE = 05 72 Q9 elements. Here exact flux used on R = 2, NBC at X = 0, EBC elsewhere ! Solution of the 2-D Poisson Equation on an segment of a circular ! annulus in the first quadrant from r=1 to r=2 ! r = 2 |-----\ ! q_n = 0..| \ .... q_n = 4 K X^2 Y^2 / R ! r = 1 |--\ K,Q \ Q = 4(X^2 + y^2) = 4 R^2 ! T=10-X^2+X^4...\ | K = 2 ! Y | | T(X,Y) = 10 - X^2 Y^2, exact ! o-X ------- ... T = 10 (10 is arbitrary constant) ! Dirchlet BC: constant at Y=0, variable at R=1 ! Normal flux: variable on R = 2, Zero on X = 0 (or on X = Y) ! Exact bc values will over-ride null input values below ! Note: on a constant radius R the normal flux is ! q_n = - 4.d0 * K * (X**2 * Y**2) / R ! For constant A angle line the normal flux is ! q_n = - 2.d0 * K * X * Y * (X * Sin A - Y * Cos A) ! PROP(1) = CONDUCTIVITY K_XX 1_D, 2_D, & 3_D ! PROP(2) = CONDUCTIVITY K_YY 2_D, & 3_D ! PROP(3) = CONDUCTIVITY K_XY, OR K_ZZ IF 3_D ! Version to substitute at QP ! Normal flux exact values substituted on input in set 3 DATA_SET = 10 EXACT_CASE = 19 Carslaw & Jaeger, Conduction of Heat in Solids, Oxford Press 1959, page 170, Kx=2, Ky=1.2337, Kxy=0, Q=1000 188 Q4 elements DATA_SET = 11 EXACT_CASE = 23 Kreyszig unit sphere with EBC The surface temperature BC is T=cosine(angle_from_Z)^2 so it varies from 1 to zero. R_max = 1 = Z_max. T_exact=(R^2 + Z^2)*( cos(ang)^2 - third) + third DATA_SET = 12 EXACT_CASE = 05 72 Q9 elements. Here essential bc is given on all edges. ! 2-D Poisson Equation on an segment of a circular from 0 to 45 deg ! annulus in the first quadrant from r=1 to r=2 ! r_outer = 2 ! q_n = / \ .... q_n = 4 X^2 Y^2 ! r_inner = 1 / \ Q = 4(X^2 + y^2) = 4 R^2 ! T=10-X^2+X^4...\ K,Q | K = 2 ! Y | | T(X,Y) = 10 - X^2 Y^2, exact ! o-X ------- ... T = 10 (10 is arbitrary constant) ! Dirchlet BC: constant at Y=0, variable at R=1 ! Normal flux: variable on R = 2, Zero on X = 0 (or on X = Y) ! ! Exact bc values will over-ride null input values ! PROP(1) = CONDUCTIVITY K_XX 1_D, 2_D, & 3_D ! PROP(2) = CONDUCTIVITY K_YY 2_D, & 3_D ! PROP(3) = CONDUCTIVITY K_XY, OR K_ZZ IF 3_D DATA_SET = 13 EXACT_CASE = 4 ! Solution of the 2-D test case used by Oden and Zienkwicz ! a square from (0,0) to (1,1) with strong diagonal gradients ! The solution is symmetric about y = x ! Comp Meth App Mech Engrg 77 (1989) 79-212 ! I J Num Meth Engrg 33 (1992) 207-224 ! The source term is complicated. C sensitive to gauss input DATA_SET = 14 EXACT_CASE = 24 ! 2D LAPLACE PATCH TEST, T6, ESSENTIAL BC METHOD ! 5--8-13-16-21 (4,4) Mesh shown to left. ! : / : / : Exact solution u = 1 + 3x - 4y ! 4 24 12 25 20 ! :/ :/ : du/dx = 3, du/dy = -4 ! 3--7-11-15-19 ! : / : / : ! 2 22 10 23 18 ! :/ :/ : ! 1--6--9-14--17 ->X DATA_SET = 15 EXACT_CASE = 20 ! Laplace eq exact cubic, 3 by 2 rectangle, EXACT_CASE 20 ! u,xx + u,yy = 0, u=-x^3 -y^3 + 3x^2y + 3xy^2 ! K_xx U,xx + 2K_xy U,xy + K_yy U,yy + Q = 0 ! Properties: K_xx, K_yy, K_xy, and if el_real >3, Q ! 482 T3 elements, like DATA_SET 08