Finite Element Method - Edge - based finite element formulation _appc This monograph presents in detail the novel "wave" approach to finite element modeling of transient processes in solids. Strong discontinuities of stress, deformation, and velocity wave fronts as well as a finite magnitude of wave propagation speed over elements are considered. These phenomena, such as explosions, shocks, and seismic waves, involve problems with a time scale near the wave propagation time. Software packages for 1D and 2D problems yield significantly better results than classical FEA, so some FORTRAN programs with the necessary comments are given in the appendix. The book is written for researchers, lecturers, and advanced students interested in problems of numerical modeling of non-stationary dynamic processes in deformable bodies and continua, and also for engineers and researchers involved designing machines and structures, in which shock, vibro-impact, and other unsteady dynamics and waves processes play a significant role.
Trang 1Appendix C
Edge-based finite element
formulation
The edge-based data structure has been used in many recent finite element formula- tions for flow problems As mentioned in Sec 6.8, Chapter 6, this formulation has many advantages such as smaller storage, etc To explain the formulation we shall consider the Euler equations and a few assembled linear triangular elements on a two-dimensional finite element mesh as shown in Fig C 1 From Eq (1.24) we rewrite
the following Euler equations
where @ are the conservative variables If the element-based formulation for the above equation omits the stabilization terms, the weak form can be written as
A@
In a fully explicit form of solution procedure, the left-hand side becomes M(A@/At) and here M is the consistent mass matrix (see Chapter 3) We can write the RHS of
the above equation for an interior node I (Fig C.l(a)) by interpolating F; in each
element and after applying Green's theorem as
where AE is the area and I, J and K are the three nodes of the element (triangle) E
This is an acceptable added approximation which is frequently used in the Taylor- Galerkin method (see Chapter 2) In another form, the above RHS can be written
as (Fig C.l(a))
3 a N ~ ax; (F, + F, + F ~ ) + - 3 - ax; (F, + F~ + F,) + - 3 ~ ax j (F, + F, + F ~ ) (c.4)
where A I , A2 and A 3 are the areas of elements 1, 2 and 3 respectively For integra-
tion over the boundary on the RHS, we can write the following in the element formulation
jr N ' ( N k F f ) d r n B = [ % (2F: + Fi).]
B € I B B E 1 B
Trang 2Appendix C 299
Fig C.l Typical patch of linear triangular elements: (a) inside node; (b) boundary node
where n is the boundary normal The above equation can be rewritten for the node I
in Fig C 1 (b) as
(C.6)
~ (2Fj + F?)n, + ~ (2Fj + Ff)n2
where r B ] and r B 2 are appropriate edge lengths
The above equations (C.3) and (C.5) can be reformulated for an edge-based data
structure In such a procedure, Eq (C.3) can be rewritten as (for an interior node I)
A E 'lvl (F! + F i s )
where m S is the number of edges in the mesh which are directly connected to the node
I and the summation C E E I l , extends over those elements that contain the edges I [ T
The user can readily verify that the above equation is identically equal to the standard
element formulation of Eq (C.4) if we consider the node I in Fig C.l(a) The
inclusion of boundary sides is direct from Eqs (C.5) and (C.6)