For convenience of later expression, for any subdomain (Asubseteq Omega), we introduce the following energy norm (|v|_{varepsilon(A)}): |v|^{2}_{varepsilon(A)}=B_{A} v,v),quad forall vin H^{1}(A), where (B_{A} u,w):=int_{A} anabla ucdotnabla w,dx,dy) for any (u,win H^{1}(A)).
For the convenience of expression in the rest of the paper, we let (mu_{0}=0).
For the convenience of expression, some abbreviations are given as follows.
For the convenience of expression, we assume the macro-BS and pico-BS as BS 1 and BS 2, respectively.
For the convenience of expression in rest of the paper, we let μ 0 = 0. Theorem 3.1 Suppose f ( t, 0, …, 0 ) ≢ 0 for any t ∈ [ 0, 1 ].
For the convenience of expression, the equations in (18) are transformed into the equation in (19), where J ac is the Jacobi matrix of the correction equations.
Thus, the overall complexity of the ET-FSD is dominated by p as ζ p =O(m p ), where ζ p, for the convenience of expression, is defined as the overall complexity of the ET-FSD when the FE search stage has p levels.
For convenience of engineering applications, the explicit expressions of the effective elastic moduli for composites with distributions of unidirectionally aligned, three- and two-dimension randomly oriented spheroidal particles, respectively, are obtained.
The related expressions of M T model on CTEs are summed up under self-consistent coordinate systems for convenience of practical use.
