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In recent years, cellular materials have been widely studied and applied in aerospace and other fields due to the advantages of lightweight and multi-function. However, it is difficult to predict the equivalent elastic properties of the graded lattice structure and other non-uniform cellular materials because of the complex configuration and non-uniformity. A new discretization method for predicting the equivalent elastic parameters of the graded lattice structure is proposed based on the strain energy equivalent method and the discretization method in this paper. The graded lattice structure is discretized into lattice cells, the equivalent elastic properties are predicted by calculating the global equivalent elastic parameters with the parameters of lattice cells, and the calculation formulas are derived. After that, taking edge cube, face-centered cubic and body-centered cubic lattice as examples, the effectiveness and accuracy of the method are verified by theoretical calculation, numerical analysis, and experiment. The results show that the calculation errors of equivalent elastic parameters are between 4.5%–9.7%, and the errors can be significantly improved by reducing the graded factor. It proves that the proposed discretization method can predict the equivalent elastic parameters of the graded lattice structure effectively, and is suitable for different lattice structures.

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Let g g be a holomorphic Hecke newform of level D D and λ g ( n ) \lambda _{g}(n) be its n n -th Fourier coefficient. We prove that the sum S D ( N , α , β , X ) = ∑ X > n ≤ 2 X n ≡ l \textrm {mod} N λ g ( n ) e ( α n β ) \mathcal {S}_{D}(N,\alpha , \beta ,X)=\sum _{\substack {X>n\leq 2X\\ n\equiv l \,\text {\textrm {mod}} N}}\lambda _{g}(n)e(\alpha n^\beta ) has an asymptotic formula for the case of β = 1 / 2 \beta =1/2 , α \alpha close to ± 2 q / c 2 D 2 \pm 2\sqrt {q/c^2D_2} , where l l , q q , c c , D 2 D_2 are positive integers satisfying ( l , N ) = 1 (l,N)=1 , c | N c|N , D 2 = D / ( c , D ) D_2=D/(c, D) and X X is sufficiently large. We obtain upper bounds of S D ( N , α , β , X ) \mathcal {S}_{D}(N,\alpha , \beta ,X) for the case of 0 > β > 1 0>\beta >1 and α ∈ R \alpha \in \mathbb {R} .