Matko Ljulj

University of Zagreb · Department of Mathematics

In this paper, we consider the equilibrium problem of interaction of three elastic bodies of different elastic properties. The main body is the unit cube. On top of it, there is a thin layer/cuboid of thickness ε$$ \varepsilon $$ of material whose stiffness is of order 1ε$$ \frac{1}{\varepsilon } $$ that in the middle contains another cuboid which...

In this paper we consider the equilibrium problem of interaction of three elastic bodies of different elastic properties. The main body is the unit cube. On top of it is a thin layer/quboid of thickness $\eps$ of material whose stiffness is of order $\frac{1}{\eps}$ that in the middle contains another cuboid which is of width and thickness $\eps$ t...

In this paper we consider the homogenization of a time-dependent heat conduction problem on a planar one-dimensional periodic structure. On the edges of a graph the one-dimensional heat equation is posed, while the Kirchhoff junction condition is applied at all (inner) vertices. Using the two-scale convergence adapted to homogenization of lower-dim...

In this manuscript we present a mathematical theory and a computational algorithm to study optimal design of mesh-like structures such as metallic stents by changing the stent strut thickness and width to optimize the overall stent compliance. The mathematical constrained optimization problem is to minimize the “compliance functional” over a closed...

In this article, we explore the possibility of modeling the interaction of a 2d elastic body with a thin 2d elastic body of possibly higher thickness using a 1d model for the thin body. We use the asymptotic analysis with respect to the small thickness of the 2d interaction model and formulate five different limit models depending on the order of s...

The standard engineer's model for heat transfer between the fluid flowing through the pipe and the exterior medium neglects the effects of the pipe's wall. The goal of this paper is to prove that they are not always negligible. Comparing the ratio between diffusivities of the fluid and the wall with the wall's thickness, using rigorous asymptotic a...

A new model description for the numerical simulation of elastic frame structures is proposed. Instead of resolving algebraic constraints at frame nodes and incorporating them into the finite element spaces, the constraints are included explicitly in the model via new variables and enforced via Lagrange multipliers. Based on the new formulation, an...

In this paper a new nonlinear two-dimensional shell model of Naghdi’s type is formulated for shells which middle surface is parameterized by a \(W^{1, \infty}\) function. Therefore the model inherently contains undeformed geometries with corners, so the model also includes models of junctions of nonlinear shells. Deformation of the shell is describ...

In this paper we consider the homogenization of a time-dependent heat conduction problem on a planar one-dimensional periodic structure. On the edges of a graph the one-dimensional heat equation is posed, while the Kirchhoff junction condition is applied at all (inner) vertices. Using the two-scale convergence adapted to homogenization of lower-dim...

In this paper, we derive models for the interaction of a linearized three-dimensional elastic structure with a thin elastic layer of possibly different material attached to it. Rigorous derivation is performed by considering a thin three-dimensional layer and the asymptotics of the solution of the full remaining three-dimensional problem when the t...

We study a linear fluid–structure interaction problem between an incompressible, viscous 3D fluid flow, a 2D linearly elastic Koiter shell, and an elastic 1D net of curved rods. This problem is motivated by studying fluid–structure interaction between blood flow through coronary arteries treated with metallic mesh-like devices called stents. The fl...

A new model description for the numerical simulation of elastic stents is proposed. Based on the new formulation an inf-sup inequality for the finite element discretization is proved and the proof of the inf-sup inequality for the continuous problem is simplified. The new formulation also leads to faster simulation times despite an increased number...

In this paper, we first formulate a linear quasi-static poroelastic shell model of Naghdi's type. The model is given in three unknowns: displacement of the middle surface, infinitesimal rotation of the cross section of the shell, and the pressure π. The model has the structure of the quasi-static Biot's system and can be seen as a system of the she...

We formulate a new free-boundary type mathematical model describing the interaction between a shell and a mesh-like structure consisting of thin rods. Composite structures of this type arise in many applications. One example is the interaction between vascular walls treated with vascular devices called stents. The new model embodies two-way couplin...