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# 유한요소법(FEM)을 이용한 구조해석

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### Transcript

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