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유한요소법(FEM)을 이용한 구조해석
유한요소법(FEM)을 이용한 구조해석
유한요소법(FEM)을 이용한 구조해석
유한요소법(FEM)을 이용한 구조해석
유한요소법(FEM)을 이용한 구조해석
유한요소법(FEM)을 이용한 구조해석
유한요소법(FEM)을 이용한 구조해석
유한요소법(FEM)을 이용한 구조해석
유한요소법(FEM)을 이용한 구조해석
유한요소법(FEM)을 이용한 구조해석
유한요소법(FEM)을 이용한 구조해석
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유한요소법(FEM)을 이용한 구조해석

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  • 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
      • Indirect addressing
      • 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
    • Conjugate Gradient Method
    Broadcast  Broadcast 
  • 10. Parallel Iterative Solver
    • Vector operations (inner product and addition of vectors) are parallelized just by distributing data
    • Computation of K x (Most of computing time spent)
      • Interface values are summed up via
      • communication
      • Interface DOFs must be minimized to
      • reduce parallel overhead
    Efficient mesh partitioning scheme required
  • 11. Paralle Explicit Time Integration
    • Central difference method
      • Compute K u ( t ) in each domain
      • Resulting K u ( t ) on interface summed up via communication
      • Compute remaining terms in each domain and obtain u ( t +  t )

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