Discrete modeling of fiber reinforced composites using the scaled boundary finite element method

A numerical method for the discrete modeling of fiber reinforced composites based on the scaled boundary finite element method (SBFEM) is proposed. A unique feature of this method is that the meshes of the matrix, aggregates, in general volumetric entities can be generated independently of the fibers which are treated as truss elements. To this end, a novel embedding method is developed which connects the mesh of the matrix consisting of scaled boundary polytopes to the fibers. This approach ensures that conforming matrix and fiber meshes are achieved. The computed stiffness matrices for both components are then simply superimposed using the nodal connectivity data. Since volume elements can be intersected by fibers at arbitrary locations, it is of paramount importance to be able to generate polytopal elements which is one unique feature of the chosen SBFEM implementation. An advantage of this procedure is that no interface constraints or special elements are required for the coupling. Furthermore, it is possible to account for random fiber distributions in the numerical analysis. In this contribution, a perfect bonding between the matrix and fibers is assumed. By means of several numerical examples, the versatility and robustness of the proposed method are demonstrated

Three-dimensional image-based numerical homogenisation using octree meshes

The determination of effective material properties of composites based on a three-dimensional representative volume element (RVE) is considered in this paper. The material variation in the RVE is defined based on the colour intensity in each voxel of an image which can be obtained from imaging techniques such as X-ray computed tomography (XCT) scans. The RVE is converted into a numerical model using hierarchical meshing based on octree decompositions. Each octree cell in the mesh is modelled as a scaled boundary polyhedral element, which only requires a surface discretisation on the polyhedron's boundary. The problem of hanging (incompatible) nodes – typically encountered when using the finite element method in conjunction with octree meshes – is circumvented by employing special transition elements. Two different types of boundary conditions (BCs) are used to obtain the homogenised material properties of various samples. The numerical results confirm that periodic BCs provide a better agreement with previously published results. The reason is attributed to the fact that the model based on the periodic BCs is not over-constrained as is the case for uniform displacement BCs

A time-domain approach for the simulation of three-dimensional seismic wave propagation using the scaled boundary finite element method

A direct time-domain approach to simulate seismic wave propagation in three-dimensional unbounded media is proposed based on the Scaled Boundary Finite Element Method (SBFEM). A domain of interest is commonly partitioned into a far field and a near field. The far field is modelled by the semi-analytical SBFEM satisfying rigorously the radiation conditions at infinity. Separate scaled boundary finite elements are employed to reach a balance between computational efficiency and accuracy. The near field is discretized into arbitrarily-shaped scaled boundary finite elements without the occurrence of hanging nodes. This advantage of the SBFEM in mesh generation is leveraged by incorporating the automatic octree-based meshing technique. By exploiting the geometrical similarity of both bounded and unbounded SBFE subdomains the computational cost is reduced. Inspired by the Domain Reduction Method (DRM), seismic waves are introduced to the system via a single layer of elements in the near field. This formulation of seismic input is mathematically convenient as it avoids the direct participation of the formulation of the far field. The proposed approach is attractive in a reliable simulation of the far field, flexible mesh generation of the near field and simple formulation of the seismic excitations. These merits are demonstrated through numerical simulations of seismic wave propagation in a free field and different examples featuring complex geometries in the near fields

Provably weak instances of ring-LWE revisited

In CRYPTO 2015, Elias, Lauter, Ozman and Stange described an attack on the non-dual decision version of the ring learning with errors problem (RLWE) for two special families of defining polynomials, whose construction depends on the modulus q that is being used. For particularly chosen error parameters, they managed to solve non-dual decision RLWE given 20 samples, with a success rate ranging from 10% to 80%. In this paper we show how to solve the search version for the same families and error parameters, using only 7 samples with a success rate of 100%. Moreover our attack works for every modulus q instead of the q that was used to construct the defining polynomial. The attack is based on the observation that the RLWE error distribution for these families of polynomials is very skewed in the directions of the polynomial basis. For the parameters chosen by Elias et al. the smallest errors are negligible and simple linear algebra suffices to recover the secret. But enlarging the error paremeters makes the largest errors wrap around, thereby turning the RLWE problem unsuitable for cryptographic applications. These observations also apply to dual RLWE, but do not contradict the seminal work by Lyubashevsky, Peikert and Regev

Relating different Polynomial-LWE problems

In this paper we focus on Polynomial Learning with Errors (PLWE). This problem is parametrized by a polynomial and we are interested in relating the hardness of the $\text{PLWE}^f$ and $\text{PLWE}^h$ problems for different polynomials $f$ and $h$. More precisely, our main result shows that for a fixed monic polynomial $f$, $\text{PLWE}^{f\circ g}$ is at least as hard as $\text{PLWE}^f$, in both search and decision variants, for any monic polynomial $g$. As a consequence, $\text{PLWE}^{\phi_n}$ is harder than $\text{PLWE}^{f},$ for a minimal polynomial $f$ of an algebraic integer from the cyclotomic field $\mathbb{Q}(\zeta_n)$ with specific properties. Moreover, we prove in decision variant that in the case of power-of-2 polynomials, $\text{PLWE}^{\phi_n}$ is at least as hard as $\text{PLWE}^f,$ for a minimal polynomial $f$ of algebraic integers from the $n$th cyclotomic field with weaker specifications than those from the previous result

A Subfield Lattice Attack on Overstretched NTRU Assumptions:Cryptanalysis of Some FHE and Graded Encoding Schemes

International audienc

On the Hardness of the Computational Ring-LWR Problem and its Applications

In this paper, we propose a new assumption, the Computational Learning With Rounding over rings, which is inspired by the computational Diffie-Hellman problem. Assuming the hardness of ring-LWE, we prove this problem is hard when the secret is small, uniform and invertible. From a theoretical point of view, we give examples of a key exchange scheme and a public key encryption scheme, and prove the worst-case hardness for both schemes with the help of a random oracle. Our result improves both speed, as a result of not requiring Gaussian secret or noise, and size, as a result of rounding. In practice, our result suggests that decisional ring-LWR based schemes, such as Saber, Round2 and Lizard, which are among the most efficient solutions to the NIST post-quantum cryptography competition,stem from a provable secure design. There are no hardness results on the decisional ring-LWR with polynomial modulus prior to this work, to the best of our knowledge