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This paper develops an extended bi-directional evolutionary structural optimization (BESO) method for topology optimization of continuum structures with smoothed boundary representation. In contrast to conventional zigzag BESO designs and removal/addition of elements, the newly proposed evolutionary topology optimization (ETO) method, determines implicitly the smooth structural topology by a level-set function (LSF) constructed by nodal sensitivity numbers. The projection relationship between the design model and the finite element analysis (FEA) model is established. The analysis of the design model is replaced by the FEA model with various elemental volume fractions, which are determined by the auxiliary LSF. The introduction of sensitivity LSF results in intermediate volume elements along the solid-void interface of the FEA model, thus contributing to the better convergence of the optimized topology for the design model. The effectiveness and robustness of the proposed method are verified by a series of 2D and 3D topology optimization design problems including compliance minimization and natural frequency maximization. It has been shown that the developed ETO method is capable of generating a clear and smooth boundary representation; meanwhile the resultant designs are less dependent on the initial guess design and the finite element mesh resolution.

The traditional topology optimization method of continuum structure generally uses quadrilateral elements as the basic mesh. This approach often leads to jagged boundary issues, which are traditionally addressed through post-processing, potentially altering the mechanical properties of the optimized structure. A topology optimization method of Movable Morphable Smooth Boundary (MMSB) is proposed based on the idea of mesh adaptation to solve the problem of jagged boundaries and the influence of post-processing. Based on the ICM method, the rational fraction function is introduced as the filtering function, and a topology optimization model with the minimum weight as the objective and the displacement as the constraint is established. A triangular mesh is utilized as the base mesh in this method. The mesh is re-divided in the optimization process based on the contour line, and a smooth boundary parallel to the contour line is obtained. Numerical examples demonstrate that the MMSB method effectively resolves the jagged boundary issues, leading to enhanced structural performance.

This study presents an extension of the Smooth-Edged Material Distribution Optimisation Technique (SEMDOT) to multiscale topology optimisation (MSTO). While the SEMDOT has shown promise in producing smooth and fabrication-friendly structures in various single-scale problems, its application to multiscale design remains unexplored. To extend SEMDOT to MSTO, a discrete sensitivity approach without penalisation is introduced, in which sensitivities are directly determined by classifying elements. Microstructural properties are computed using energy-based homogenisation with periodic boundary conditions (PBCs), enabling efficient and accurate prediction of effective elastic moduli. Physical fidelity of the smooth boundaries estimated by level-set functions are validated. Numerical results from 2D and 3D compliance minimization benchmarks demonstrate the effectiveness of the SEMDOT method, resulting in smooth boundaries between solid and void phases at both macro- and microscales, overcoming the jagged boundaries and grayscale issues seen in conventional methods. The results also show that the SEMDOT achieves comparable performance to other MSTO methods, with fewer iterations and reduced computational time.

The goal of this work is to construct the general solution of the Cauchy–Riemann equation with strong singularities in the lower order coefficient and to study the Riemann–Hilbert boundary value problem in a domain with a piecewise smooth boundary.

In this paper, we introduce an m -fold illumination number I m ( K ) of a convex body K in Euclidean space E d , which is the smallest number of directions required to m -fold illuminate K , i.e., each point on the boundary of K is illuminated by at least m directions. We get a lower bound of I m ( K ) for any d -dimensional convex body K , and get an upper bound of I m ( K ) for any d -dimensional convex body K with smooth boundary. We also prove that I m ( K ) = 2 m + 1 , for a 2-dimensional smooth convex body K . Furthermore, we obtain some results related to the m -fold illumination numbers of convex polygons and cap bodies of a d -dimensional unit ball B d in small dimensions. In particular, we show that , for a regular convex n -sided polygon P .

This paper proposes a new lattice smooth variable structure filter (LSVSF) for maneuvering target tracking with model uncertainty. Under the assumption that the probability density function (PDF) is Gaussian-distributed, the nonlinear smooth variable structure filter (SVSF) framework is reconstructed by Bayesian theory. The optimal smooth boundary layer (OSBL) calculation form of the SVSF in a nonlinear framework is proposed. Then, based on lattice sampling methods with low computational complexity, the LSVSF algorithm is obtained. Finally, the LSVSF algorithm is verified on the maneuvering target tracking problem with model uncertainty by three scenarios: uniform motion (UM), coordinated turn (CT) motion and mixed motion (UM and CT). According to the simulation, the proposed LSVSF algorithm has superior tracking accuracy and robustness.

In this paper we consider complete noncompact Riemannian manifolds (M, g) with nonnegative Ricci curvature and Euclidean volume growth, of dimension n≥3. For every bounded open subset Ω⊂M with smooth boundary, we prove that ∫∂ΩHn-1n-1dσ≥AVR(g)|Sn-1|,where H is the mean curvature of ∂Ω and AVR(g) is the asymptotic volume ratio of (M, g). Moreover, the equality holds true if and only if (M\\Ω,g) is isometric to a truncated cone over ∂Ω. An optimal version of Huisken’s Isoperimetric Inequality for 3-manifolds is obtained using this result. Finally, exploiting a natural extension of our techniques to the case of parabolic manifolds, we also deduce an enhanced version of Kasue’s non existence result for closed minimal hypersurfaces in manifolds with nonnegative Ricci curvature.

In this paper we consider non-intersecting Brownian bridges, under fairly general upper and lower boundaries, and starting and ending data. Under the assumption that these boundary data induce a smooth limit shape (without empty facets), we establish two results. The first is a nearly optimal concentration bound for the Brownian bridges in this model. The second is that the bulk local statistics of these bridges along any fixed time converge to the sine process.

Let X ⊂ R 4 be a convex domain with smooth boundary Y . We use a relation between the extrinsic curvature of Y and the Ruelle invariant of the Reeb flow on Y to prove that there are constants C > c > 0 independent of Y such that c ⩽ ru ( Y ) · sys ( Y ) 1 / 2 ⩽ C Here sys ( Y ) is the systolic ratio of Y , i.e. the square of the minimal period of a closed Reeb orbit of Y divided by twice the volume of X , and ru ( Y ) is the volume-normalized Ruelle invariant. We then construct dynamically convex contact forms on S 3 that violate this bound using methods of Abbondandolo–Bramham–Hryniewicz–Salomão. These are the first examples of dynamically convex contact 3-spheres that are not strictly contactomorphic to a convex boundary Y .

We consider the following problem: -Δpu=c(x)|u|q-1u+μ|∇u|p+h(x) in Ω, u=0 on ∂Ω, where Ω is a bounded set in RN (N≥3) with a smooth boundary, 10, μ∈R⁎, and c and h belong to Lk(Ω) for some k>N/p. In this paper, we assume that c≩0 a.e. in Ω and h without sign condition and then we prove the existence of at least two bounded solutions under the condition that ck and hk are suitably small. For this purpose, we use the Mountain Pass theorem, on an equivalent problem to (P) with variational structure. Here, the main difficulty is that the nonlinearity term considered does not satisfy Ambrosetti and Rabinowitz condition. The key idea is to replace the former condition by the nonquadraticity condition at infinity.