A sharp interface method using enriched finite elements for elliptic interface problems

We present an immersed boundary method for the solution of elliptic interface problems with discontinuous coefficients which provides a second-order approximation of the solution. The proposed method can be categorised as an extended or enriched finite element method. In contrast to other extended FEM approaches, the new shape functions get projected in order to satisfy the Kronecker-delta property with respect to the interface. The resulting combination of projection and restriction was already derived in Höllbacher and Wittum (TBA, 2019a) for application to particulate flows. The crucial benefits are the preservation of the symmetry and positive definiteness of the continuous bilinear operator. Besides, no additional stabilisation terms are necessary. Furthermore, since our enrichment can be interpreted as adaptive mesh refinement, the standard integration schemes can be applied on the cut elements. Finally, small cut elements do not impair the condition of the scheme and we propose a simple procedure to ensure good conditioning independent of the location of the interface. The stability and convergence of the solution will be proven and the numerical tests demonstrate optimal order of convergence.

Adaptive 3D unstructured mesh refinement

Algorithms for generating three-dimensional detailed computational meshes are considered. The algorithms are based on adaptive refinement of the original coarse meshes describing a 3D object. The purpose of adaptation is to form an accurate description of the volume and surface of a three-dimensional object for supercomputer modeling. Refinement of the boundary description is performed by projecting the cut elements of the coarse mesh onto the corresponding elements of the object’s surface.

FRICTION FORCE OF THE NATURAL LEATHER CUTTING PROCESS

Cutting natural leather is a first stage of a technological process of leather goods. It consists of dividing flat materials into shoe components. Cutting can be done by hand or mechanically. Cutters are the main cutting device in mechanical process. During utilization of cutters, the durability of its usage is an important issue. To increase its endurance, optimal force conditions are required. These depend on the following: the friction coefficient between cutting machine and cut material, the angle of sharpening of cutter, the blunting of the edge, using a cutting pad, and the type and kind of steel of which is produced. Friction is a component of a cutting force and its value depends on several factors, one of which is shape of the cutter. It is a parameter often ignored in the literature; however, according to the studies conducted by the authors, the value of the friction during cutting depends on the shape of the cutter. In this study, results of force value measurements required to cut elements of different shapes with equal circumferences with steel cutters are presented.

Splitting schemes and unfitted-mesh methods for the coupling of an incompressible fluid with a thin-walled structure

Two unfitted-mesh methods for a linear incompressible fluid/thin-walled structure interaction problem are introduced and analyzed. The spatial discretization is based on different variants of Nitsche’s method with cut elements. The degree of fluid–solid splitting (semi-implicit or explicit) is given by the order in which the space and time discretizations are performed. The a priori stability and error analysis shows that strong coupling is avoided without compromising stability and accuracy. Numerical experiments with a benchmark illustrate the accuracy of the different methods proposed.