유한요소법 (FEM) 을 이용한 구조해석
구조해석 기법의 분류 <ul><li>물리적인 대상 기준  : formulation procedure ( 유한요소법 사용 ) </li></ul><ul><ul><li>정적해석  (Static analysis) :  선형 ,...
수치기법별 특성 <ul><li>내재적 기법  : sparse matrix 로 구성된 연립방정식을 풀어야 함 </li></ul><ul><ul><li>Sparse matrix solver </li></ul></ul><ul>...
<ul><li>Direct solver for Implicit method </li></ul><ul><ul><li>Profile solver (Band solver, Skyline solver) : Node renumb...
Solution procedure for Structural Analysis <ul><li>Compute  M i (e) ,  K i (e) </li></ul><ul><ul><li>How to formulate the ...
Parallel Direct Solver <ul><li>Multifrontal Solver (direct solver) </li></ul><ul><ul><li>Data redistribution in extend-add...
Parallel Direct Solver <ul><li>Distributed memory parallelization of multifrontal solver </li></ul><ul><ul><li>Parallel ex...
Parallel Direct Solver <ul><li>Distributed memory parallelization of multifrontal solver </li></ul><ul><ul><li>Parallel ex...
Parallel Iterative Solver <ul><li>Element-wise data distribution also valid for solution procedure </li></ul><ul><li>Conju...
Parallel Iterative Solver <ul><li>Vector operations (inner product and addition of vectors)  are parallelized just by dist...
Paralle Explicit Time Integration <ul><li>Central difference method  </li></ul><ul><ul><li>Compute  K u ( t )  in each dom...
Upcoming SlideShare
Loading in …5
×

유한요소법(FEM)을 이용한 구조해석

6,028 views

Published on

Published in: Education, Technology
0 Comments
2 Likes
Statistics
Notes
  • Be the first to comment

No Downloads
Views
Total views
6,028
On SlideShare
0
From Embeds
0
Number of Embeds
53
Actions
Shares
0
Downloads
0
Comments
0
Likes
2
Embeds 0
No embeds

No notes for slide

유한요소법(FEM)을 이용한 구조해석

  1. 1. 유한요소법 (FEM) 을 이용한 구조해석
  2. 2. 구조해석 기법의 분류 <ul><li>물리적인 대상 기준 : formulation procedure ( 유한요소법 사용 ) </li></ul><ul><ul><li>정적해석 (Static analysis) : 선형 , 비선형 </li></ul></ul><ul><ul><li>Ku = f </li></ul></ul><ul><ul><li>동적해석 (Dynamic analysis) : 선형 , 비선형 </li></ul></ul><ul><ul><ul><li>time response </li></ul></ul></ul><ul><ul><ul><li>Mü + Cú + Ku = f </li></ul></ul></ul><ul><ul><ul><li>frequency response </li></ul></ul></ul><ul><ul><ul><li>(  M - K ) x =0 </li></ul></ul></ul><ul><li>수치적 기법 기준 : solution procedure </li></ul><ul><ul><li>내재적 (Implicit) : stiffness 행렬 K 가 좌변에 ... </li></ul></ul><ul><ul><li>(  M +  K ) u (t+  t)= f eff (t) </li></ul></ul><ul><ul><li>외연적 (Explicit) </li></ul></ul><ul><ul><li>  M u (t+  t)= f eff (t) </li></ul></ul>
  3. 3. 수치기법별 특성 <ul><li>내재적 기법 : sparse matrix 로 구성된 연립방정식을 풀어야 함 </li></ul><ul><ul><li>Sparse matrix solver </li></ul></ul><ul><ul><ul><li>Direct solver </li></ul></ul></ul><ul><ul><ul><ul><li>out-of-core solution </li></ul></ul></ul></ul><ul><ul><ul><ul><li>multiple RHS </li></ul></ul></ul></ul><ul><ul><ul><ul><li>병렬 계산 시 복잡하고 많은 양의 통신이 발생 </li></ul></ul></ul></ul><ul><ul><ul><li>Iterative solver </li></ul></ul></ul><ul><ul><ul><ul><li>multiple RHS 처리 곤란 </li></ul></ul></ul></ul><ul><ul><ul><ul><li>Conjugate gradient method - 병렬화 용이 </li></ul></ul></ul></ul><ul><ul><li>ABAQUS/Standard(direct/shared), NASTRAN(direct/distributed) </li></ul></ul><ul><li>외연적 기법 </li></ul><ul><ul><li>행렬 - 벡터 곱으로 구성 </li></ul></ul><ul><ul><li>f eff (t) 계산 시 K u (t) 계산 ( K u =  K i (e) u i (e) ) </li></ul></ul><ul><ul><li>병렬화 용이 ( 벡터의 경계면에서의 성분에 대해서만 데이터 교환 ) </li></ul></ul><ul><ul><li>LS-DYNA, ABAQUS/Explicit </li></ul></ul>
  4. 4. <ul><li>Direct solver for Implicit method </li></ul><ul><ul><li>Profile solver (Band solver, Skyline solver) : Node renumbering </li></ul></ul><ul><ul><li>Frontal solver : Element renumbering </li></ul></ul><ul><ul><ul><ul><li>(f : frontwidth, n : total DOF) </li></ul></ul></ul></ul><ul><ul><ul><li>Memory : max(f i ) 2 +  f i = max(f i ) 2 + n * avg(f i ) </li></ul></ul></ul><ul><ul><ul><li>Computation : n*avg(f i 2 ) </li></ul></ul></ul><ul><ul><ul><li>Communication : ? </li></ul></ul></ul><ul><ul><li>Sparse solver (or multifrontal solver) </li></ul></ul><ul><li>Explicit method and Iterative solver for Implicit method </li></ul><ul><ul><li>Indirect addressing </li></ul></ul><ul><ul><li>Memory : n *  e i (e : number of nonzero entities on ith row) </li></ul></ul><ul><ul><li>Computation : (n *  e i )/iteration </li></ul></ul><ul><ul><li>Communication : O(number of interface nodes)/iteration </li></ul></ul>
  5. 5. Solution procedure for Structural Analysis <ul><li>Compute M i (e) , K i (e) </li></ul><ul><ul><li>How to formulate the K matrix in order to simulate the behavior of real-world material </li></ul></ul><ul><li>Assemble M =  M i (e) , K =  K i (e) </li></ul><ul><li>Solve </li></ul><ul><ul><li>Ku = f </li></ul></ul><ul><ul><ul><li>Implicit : solve linear or non-linear ( K depends on u ) equation </li></ul></ul></ul><ul><ul><li>Mü + Cú + Ku = f </li></ul></ul><ul><ul><ul><li>Implicit </li></ul></ul></ul><ul><ul><ul><li>(  M +  K ) u (t+  t)= f eff (t) </li></ul></ul></ul><ul><ul><ul><li>Explicit </li></ul></ul></ul><ul><ul><ul><li>  M u (t+  t)= f eff (t) </li></ul></ul></ul><ul><ul><li>(  M - K ) x =0 </li></ul></ul><ul><ul><ul><li>Eigen-value problem </li></ul></ul></ul>
  6. 6. Parallel Direct Solver <ul><li>Multifrontal Solver (direct solver) </li></ul><ul><ul><li>Data redistribution in extend-add operation </li></ul></ul><ul><ul><ul><li>PBLAS & ScaLAPACK ? : </li></ul></ul></ul><ul><ul><ul><ul><li>heavy data redistribution (all-to-all) required </li></ul></ul></ul></ul>Proc 0 Proc 1 Proc 2 Proc 3 Proc 2,3 Proc 0,1 Proc 0,1,2,3 Factorization, Forward-substitution Back- substitution Parallel BLAS & LAPACK required Factorization, Forward-substitution Back- substitution
  7. 7. Parallel Direct Solver <ul><li>Distributed memory parallelization of multifrontal solver </li></ul><ul><ul><li>Parallel extend-add operation (A. Gupta) </li></ul></ul><ul><ul><ul><li>Row-wise & column-wise matrix partitioning is repeated in turn </li></ul></ul></ul><ul><ul><ul><li>Ownership of each row or column of the frontal matrix is determined by the bit pattern of its ID </li></ul></ul></ul><ul><ul><ul><li>(odd number to the left, even number to the right) </li></ul></ul></ul>+ = row-wise partitioning row-wise partitioning column-wise partitioning
  8. 8. Parallel Direct Solver <ul><li>Distributed memory parallelization of multifrontal solver </li></ul><ul><ul><li>Parallel extend-add operation (A. Gupta) </li></ul></ul><ul><ul><ul><li>Resulting communication pattern </li></ul></ul></ul><ul><ul><ul><li>PBLAS & ScaLAPACK can not be used for resulting data distribution (block size must be fixed) </li></ul></ul></ul>New parallel linear algebra subroutines are required which allow flexible block size
  9. 9. Parallel Iterative Solver <ul><li>Element-wise data distribution also valid for solution procedure </li></ul><ul><li>Conjugate Gradient Method </li></ul>Broadcast  Broadcast 
  10. 10. Parallel Iterative Solver <ul><li>Vector operations (inner product and addition of vectors) are parallelized just by distributing data </li></ul><ul><li>Computation of K x (Most of computing time spent) </li></ul><ul><ul><li>Interface values are summed up via </li></ul></ul><ul><ul><li>communication </li></ul></ul><ul><ul><li>Interface DOFs must be minimized to </li></ul></ul><ul><ul><li>reduce parallel overhead </li></ul></ul>Efficient mesh partitioning scheme required
  11. 11. Paralle Explicit Time Integration <ul><li>Central difference method </li></ul><ul><ul><li>Compute K u ( t ) in each domain </li></ul></ul><ul><ul><li>Resulting K u ( t ) on interface summed up via communication </li></ul></ul><ul><ul><li>Compute remaining terms in each domain and obtain u ( t +  t ) </li></ul></ul>

×