The main goal of the current work is to analyse the three-dimensional approach for modelling knitted fabric structures for future analysis of physical properties and thermal phenomena. The introduced model assumes some simplification of morphology. First, fibres in knitted fabrics are described as monofilaments characterized by isotropic thermal properties. The current form of the considered knitted fabric is determined by morphological properties of the used monofilament and simplification of the stitch shape. This simplification was based on a particular technology for the knitting process that introduces both geometric parameters and physical material properties. Detailed descriptions of heat transfer phenomena can also be considered. A sensitivity analysis of the temperature field with respect to selected structural parameters was also performed.

Let us next analyse the coupled problem during ironing of textiles, that is, the heat is transported with mass whereas the mass transport with heat is negligible. It is necessary to define both physical and mathematical models. Introducing two-phase system of mass sorption by fibres, the transport equations are introduced and accompanied by the set of boundary and initial conditions. Optimisation of material thickness during ironing is gradient oriented. The first-order sensitivity of an arbitrary objective functional is analysed and included in optimisation procedure. Numerical example is the thickness optimisation of different textile materials in ironing device

A steady heat conduction problem is considered, that is described by the heat conduction equation and the thermal boundary conditions (i.e., Dirichlet, Neumann, Henkel, and radiation conditions on the external boundary, and radiation condition on the hole boundary). An arbitrary behavioral functional is defined and its first-order sensitivity is derived using both the direct and the adjoint approaches. The shape optimization problem is next formulated and two optimization functionals are discussed. The simple numerical example is presented.

A steady heat conduction problem within an anisotropic domain is formulated. Considering an arbitrary behavioral functional, its first and second sensitivities with respect to variation of domain shape are derived using the material derivative concept and direct as well as adjoint approaches. Then the shape optimal design problem is formulated and the optimization algorithms of the first and second order are discussed.