Remaining ASEN 5007 Class Schedule Today Solving FEM Equations - PDF document

Introduction to FEM Remaining ASEN 5007 Class Schedule Today Solving FEM Equations (descriptive, no assignments) 12/7 Demo of a complete plane stress program (Ch 27) 12/9 Description of take-home exam Stress recovery (Ch 28) Last HW (#11) due Dec 9 (on-campus) Take-home exam posted Dec 8, also posted on web site that day

Introduction to FEM 26 Solving FEM Equations IFEM Ch 26 – Slide 1

Introduction to FEM Role of the Solver in a FEM Code Model definition tables: ^ geometry K K Modify Eqs Equation Assembler element Solver for BCs freedom connectivity Some equation solvers apply BCs loop over and solve simultaneously (e) K elements Nodal Element displacements Stiffness Matrices ELEMENT LIBRARY IFEM Ch 26 – Slide 2

Introduction to FEM Computer Resources Req'd by FEM Solver Storage and Solution Times for a Fully Stored Stiffness Matrix Matrix Storage Factor Factor time Factor time order N (double prec) op. units workstation/PC supercomputer 10 4 10 12 / 6 2 min 800 MB 3 hrs 10 5 10 15 / 6 30 hrs 80 GB 4 mos 10 6 10 18 / 6 300 yrs 3 yrs 8 TB time numbers last adjusted in 1998 to get current times divide by 10-20 IFEM Ch 26 – Slide 3

Introduction to FEM Typical Stiffness Matrix Sparsity Pattern N = 50 y F 1 6 11 16 21 (1) (5) (9) (13) 2 12 17 22 7 (2) (6) (10) (14) 3 8 13 18 23 K = x (3) (7) (11) (15) 4 9 14 19 24 (4) (8) (12) (16) 5 10 15 20 25 IFEM Ch 26 – Slide 4

Introduction to FEM Computer Resources Req'd by FEM Solver Storage and Solution Times for a Skyline Stiffness Matrix √ Assuming B = N Matrix Storage Factor Factor time Factor time order N (double prec) op. units workstation/PC supercomputer 10 4 10 8 / 2 5 sec 8 MB 0.05 sec 10 5 10 10 / 2 8 min 240 MB 5 sec 10 6 10 12 / 2 15 hrs 8 GB 8 min time numbers last adjusted in 1998 to get current times divide by 10-20 IFEM Ch 26 – Slide 5

Introduction to FEM What We Will Cover Today How the Master Stiffness Equations are stored in a commonly used "skyline" sparse format How to Mark BC on the Master Stiffness Eqs (if you write your own solver) The Basic Solution Steps Note: Implementation Details will be Skipped since Built-in Mathematica Solver will be used for Demo Programs IFEM Ch 26 – Slide 6

Introduction to FEM Skyline Storage (aka Profile or Variable-Band Storage)   11 13 16 Skyline 22 0 24 0     "template" 33 34 0  K =    44 46     55 56   symm 66 Mathematica representation p = { 0, 1, 2, 5, 8, 15}; s = {11, 22, 13, 0, 33, 24, 34, 44, 55, 16, 0, 0, 46, 66}; S = { p, s}; IFEM Ch 26 – Slide 7

Introduction to FEM Marking Displacement BCs Equations for which the displacement component is known or prescribed are identified by a negative diagonal location value. For example, if u and u are prescribed 5 3 displacement components in the example system, p : [ 0 , 1 , 2 , − 5 , 8 , − 9 , 15 ] IFEM Ch 26 – Slide 8

Introduction to FEM Solution Steps for K u = f Factorization T K = L D U = L D L Solution Forward reduction: solve L z = f for z Diagonal Scaling: solve D y = z for y Back substitution: solve U u = y for u IFEM Ch 26 – Slide 9

Introduction to FEM SkySolver Implementation (sketched only, Notes give details) S = { p, s }  11 13 16  22 0 24 0     33 34 0   K =   44 46     55 56   symm 66 IFEM Ch 26 – Slide 10

Introduction to FEM SkySolver Implementation (cont'd) Mathematica representation of skyline array p = { 0, 1, 2, 5, 8, 15}; s = {11, 22, 13, 0, 33, 24, 34, 44, 55, 16, 0, 0, 46, 66}; S = { p, s}; In one list S = {{ 0,1,2,5,8,15} , {11,22,13,0,33,24,34,44,55,16,0,0,46,66}}; Further implementation details given in Chapter 26 of Notes in sections marked *, but those are intended for a more advanced course IFEM Ch 26 – Slide 11