For solving a partial different equation by a numerical method, a possible alternative may be either to use a mesh method or a meshless method. A flexible computational procedure for solving 1D linear elastic beam problems is presented that currently uses two forms of approximation function (moving least squares and kernel approximation functions) and two types of formulations, namely the weak form and collocation technique, respectively, to reproduce Element Free Galerkin (EFG) and Smooth Particle Hydrodynamics (SPH) meshless methods. The numerical implementation for beam problems of these two formulations is discussed and numerical tests are presented to illustrate the difference between the formulations.

The paper describes anatomical and physiological features of photobionts and mycobionts in Bryoria forsteri Olech & Bystrek, Caloplaca regalis (Vain.) Zahlbr., Cetraria aculeata (Schreb.) Fr., Ramalina terebrata Hook f. & Taylor, Sphaerophorus globosus (Huds.) Vain. and Usnea antarctica Du Rietz, collected in the Antarctic under varied weather conditions. Green algae from the genera Lobosphaera and Trebouxia were gathered in depressions of the cortex under the more resistant mycobiont hyphae. In photobiont cells a large amount of highly osmiophilic electron-dense PAS-negative material, lipid-like in character, was of particular interest. Similar material also filled certain areas of the aerial apoplast. A star-shaped chromatophore with central and lateral pyrenoids encompassed most of the photobiont protoplast in all the studied species. Regularly arranged thylakoids with evenly widened lumina along their entire length and osmiophilic lipid droplets adhering to their outer surfaces were visible within the pyrenoid. Inside the chloroplast, large protein inclusions tightly joined with the thylakoids were observed. The mycobionts were closely attached to each other another and with the photobionts by means of an outer osmiophilic wall layer, and formed intramural haustoria. Their protoplasts were filled with PAS-positive polysaccharides and a large amount of lipid-like substances. The photobionts were physiologically active and produced a large amount of electron-dense osmiophilic material, and PAS-positive starch grains were visible around their pyrenoids in the thalli collected in different weather conditions. The permanent reserves of nutritive materials deposited in the thalli enable these organisms to quickly begin and continue indispensable physiological processes in the extreme Antarctic conditions.