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2 3 2 2
For example, the sphere S ⊂ R (x + y +
2
z = 1) is both a real and a complex manifold.
The real structure is given by the real charts
M x y
ϕ (x,y,z) = , ,z = 1 and
N
1 − z 1 − z

x y
macromolecule (polymer molecule) A ϕ (x,y,z) = , ,z = −1
S
1 + z 1 + z
molecule of high relative molecular mass, the
structure of which essentially comprises the while the complex atlas is given by
multiple repetition of units derived, actually or
conceptually, from molecules of low relative x + iy
ϕ (x,y,z) = ,z = 1 and
N
molecular mass. 1 − z
Notes: (1) In many cases, especially for syn- x − iy
ϕ (x,y,z) = ,z = −1.
S
thetic polymers, a molecule can be regarded as 1 + z
having a high relative molecular mass if the addi-
tion or removal of one or a few of the units has mapping See function.
a negligible effect on the molecular properties.
This statement fails in the case of certain macro- mass, m m m Base quantity in the system of
molecules for which the properties may be criti- quantities upon which SI is based.
cally dependent on fine details of the molecular
structure.
mass lumping Approximation of the mass
(2) If a part of the whole of the molecule
matrix by a diagonal matrix. This is usually
has a high relative molecular mass and essen- 2
achieved by replacing the L (;)-inner product
tially comprises the multiple repetition of
with a mesh-dependent inner product (., .) on a
h
units derived, actually or conceptually, from
finite element space V that is based on numer-
h
molecules of low relative molecular mass, it may
ical quadrature. A prominent example is sup-
be described as either macromolecular or poly-
plied by linear Lagrangian finite elements and
meric, or by polymer used adjectivally.
vertex based quadrature: on a triangular mesh
; it boils down to
h
manifold A (usually connected) topologi-
cal space M such that there exists an atlas 3
|T |
T
T
{(U ,ϕ )} α∈I , i.e., a collection such that: (u ,v ) 2 ≈ u (a )¯v (a )
h
α
i
α
h
h L (;)
h
i
3
T ∈T h i=1
(i.) {U } is an open covering of M;
α α∈I
(ii.) ϕ : U → W is a local homeomor- × u ,v ∈ V ,
h
h
h
α
α
α
m
phism onto W = ϕ (U ) ⊂ R ;
α
α
α
T
T
T
where a , a , a stand for the vertices of the
3
2
1
(iii.) ϕ ◦ ϕ α −1 : W → W are local homeo- triangle T . As the degrees of freedom for V are
β
β
α
k
morphisms of class C . h
associated with vertices of the mesh, the diagonal
The maps ϕ ◦ ϕ −1 : W → W are called structure of the resulting “lumped” mass matrix
β α α β
transition functions. The integer m is the dimen- is evident.
sion of M.A manifold M is usually assumed to
be paracompact. massmatrix Thematrixarisingfromafinite
2
If k = 0, M is called a topological (real) element discretization of the L (;)-inner prod-
n
manifold;if k =∞, it is called a differentiable uct (; ⊂ R a bounded domain): if ψ ··· ,ψ N
1
(real) manifold. If the transition functions are stands for the nodal basis of the finite element
2
analytical, then M is called an analytical (real) space V ⊂ L (;), then the corresponding mass
h
manifold.If charts (U ,ϕ ) are valued in C m matrix is given by ((ψ ,ψ ) 2 ) N . The term
α α i j L (;) i,j=1
and transition functions are, holomorphic, then mass matrix has its origin in the finite element
the manifold is called a complex manifold. analysis of problems of linear elasticity.
© 2003 by CRC Press LLC