Computational Physics

13 The finite element method for partial differential equations 13.1 Introduction When we consider a partial differential equati ...

424 The finite element method for partial differential equations The integral can be discretised by dividing up the domainDintoe ...

13.2 The Poisson equation 425 Figure 13.1. Two adjacent triangles on a square (ground plane) with a linear func- tionφ(r)shown a ...

426 The finite element method for partial differential equations P Aa Ab Ac 1 2 3 Figure 13.2. The areasAa,AbandAcfor any pointP ...

13.2 The Poisson equation 427 φa,φbandφcthese values at the corresponding vertices, the solution inside the triangle is given by ...

428 The finite element method for partial differential equations This equation is defined by a matrixkwhich is called thelocal s ...

13.3 Linear elasticity 429 In that case we apply the stiffness matrix only to those points which are in the interior of the syst ...

430 The finite element method for partial differential equations Figure 13.3. Two types of deformation, compression (left) and s ...

13.3 Linear elasticity 431 The stress plus the body forces results in the displacement. It is important that the actual value of ...

432 The finite element method for partial differential equations Just as in the case of the Laplace equation, we must find an in ...

13.3 Linear elasticity 433 Theξidepend onxandy– the relation is given inEq. (13.11). From this, and from (13.17), we find εεεe= ...

434 The finite element method for partial differential equations Figure 13.4. Deformation of a beam attached to a vertical wall, ...

13.4 Error estimators 435 first derivatives of the displacement fields). A more accurate solution would lead to continuous stres ...

436 The finite element method for partial differential equations σFEM,pis the (constant) stress in that triangle. Therefore we c ...

13.5 Local refinement 437 (a) (b) Figure 13.5. Refinement of triangular grid. (a) Two ways of partitioning a triangle – partitio ...

438 The finite element method for partial differential equations 1 2 3 4 5 6 7 8 5: 2,1,3,8,6,4,7 6: 8,5,4 Figure 13.6. The data ...

13.6 Dynamical finite element method 439 Figure 13.7. Deformed elastic beam which is attached to a vertical wall and sup- ported ...

440 The finite element method for partial differential equations Hereρis the (average) mass density on the triangle. The global ...

13.7 Concurrent coupling of length scales: FEM and MD 441 processes taking place near the fissure and far away from it, it becom ...

442 The finite element method for partial differential equations MD FEM Figure 13.8. Strip modelled partly by finite elements an ...