Computer Methods for Problem Solving

Postoji mnogo metoda za rješavanje matematičkih i ostalih problema uz pomoć računala. Mogu se uglavnom podijeliti na numeričke, simboličke i analitičke te heurističke metode u koje se može uključiti i "umjetna inteligencija". Različiti problemi zahtijevaju i različit pristup rješavanju. U sve tri grupe metoda vidi se velik napredak iz godine u godinu. Postoji i sve više hibridnih metoda, koje kombiniraju analitički, numerički i heuristički pristup i kojima su već uspješno rješeni i neki vrlo teški problemi.There are a lot of computer methods for solving mathematical and other problems. Generally, they can be classified into numerical, symbolic and analytic, as well as heuristic methods which may include "artificial intelligence". For different problems, different approaches are needed. In recent years the development of all the three groups of methods has been very fast. Owing to an increasing number of hybrid methods which combine analitic, numerical and heuristic approach, some very difficult problems have been solved successfully

Non-recursive equivalent of the conjugate gradient method without the need to restart

A simple alternative to the conjugate gradient(CG) method is presented; this method is developed as a special case of the more general iterated Ritz method (IRM) for solving a system of linear equations. This novel algorithm is not based on conjugacy, i.e. it is not necessary to maintain overall orthogonalities between various vectors from distant steps. This method is more stable than CG, and restarting techniques are not required. As in CG, only one matrix-vector multiplication is required per step with appropriate transformations. The algorithm is easily explained by energy considerations without appealing to the A-orthogonality in n-dimensional space. Finally, relaxation factor and preconditioning-like techniques can be adopted easily.Comment: 9 page

Numerical, symbolic and heuristic methods

Dan je pregled primjene računala pri čemu je težište na primjene u tehničkoj mehanici i projektiranju konstrukcija. Istaknuto je da su u primjeni računala najzastupljenije numeričke, simboličke i heurističke metode, a i metode "umjetne inteligencije". Spominju se i različiti grafički prikazi o kojima se u radu ne govori posebno. Istaknuto je da se još uvijek najviše upotrebljavaju numeričke metode ali i primjena ostalih metoda vrlo brzo raste i povećava se iz godine u godinu.An overview of computer use is given and, at that, an emphasis is placed on the use of computers in engineering mechanics and structural design. It is stressed that numerical, symbolic and heuristic methods, but also those based on "artificial intelligence", are mostly suited for computer use. Various graphical representations are also mentioned but are not separately analyzed. It is emphasized that, although numerical methods are still in leading position, an increase has been noted in the use of other methods which are gaining terrain year after year

Algoritam određivanja kontakata za metodu diskretnih elemenata

The grid based variation of the contact detection algorithm, with corresponding data structures, is described. This variation is tuned for numerical analysis of large, homogenously distributed assemblages of particles of fairly equal diameters. As an example, algorithm is used in the modelling of granular material silo filling and discharge.U radu je opisana inačica na mreži utemeljenog algoritma za nalaženje parova čestica između kojih se može pojaviti međudjelovanje kratkoga dometa, posebno prilagođena numeričkim analizama velikih sustava čestica podjednakih promjera, ravnomjerno raspodijeljenih u prostoru. Algoritam je primijenjen u modeliranju punjenja i pražnjenja silosa s granularnim materijalom

Defiences of design models for engeenering structures

Razmotreni su osnovni tipovi pogrešaka koje prate proračun konstrukcija. Upozorilo se I na one pogreške koje bi se znanjem I iskustvom mogle izbjegnuti. Posebno je upozoreno na pogreške koje nastaju pri tvorbi numeričkih modela jer se u tome dijelu kompjutorskih proračuna u praksi najviše griješi. Naglašene su I neizbježne pogreške, posebno one koje nastaju pri rješavanju velikih sustava jednadžbi. Pokušalo se I odgovoriti na pitanja kolika je pouzdanost rezultata naših proračuna.Basic types of errors occurring in the design of structures are considered. Particular attention is drawn to errors that could be avoided given appropriate knowledge and experience. A special emphasis is placed on errors occuring during creation of numerical models as it is precisely in in this part of computer design that most practical errors are made. Unavoidable errors are also presented, particulary those that occur while great equation systems are being solved. An attempt is made to provide answer to the question of the extent to which our calculation results can be considered reliable

The role of creativity and technical judgement in structural engineering

U članku se opisuje specifičan način razmišljanja inženjera konstruktora koji se pojavljuje u različitim fazama projektiranja građevine. Posebno su istaknute osobine kreativnosti i prosudbe koje često imaju ključnu ulogu u izboru ispravne konstrukcije i njezinog statičkog tretmana. Pri tome se pokušalo upozoriti na intuiciju koja bitno potiče razvoj takvih osobina. Ukratko je opisan i doprinos računala pri razmišljanju i odlučivanju uz isticanje problema koji još nisu riješeni.Specific reasoning patterns of structural engineers, that appear in various phases of the design of structures, are described in the paper. A special emphasis is placed on indispensable faculties, namely creativity and judgement, as they are often of crucial importance during selection of an appropriate structural system and during its static analysis. At that, the attention is drawn to the intuition as it significantly fosters development of such faculties. The contribution of computers in the thinking and decision-making process is briefly described, and related problems that have not as yet been solved are outlined

Calculating displacement of unsymmetrical multistory buildings exposed to seismic action

Predložena je približna metoda za proračun maksimalnih horizontalnih pomaka nesimetričnih višekatnica izloženih potresu, a nazvana je metodom ekvivalentne elastične krutosti Promatrana je višekatnica malog, srednjeg i većeg ekscentriciteta. Dobiveni pomaci uspoređuju se sa odgovarajućim rezultatima nelinearnog dinamičkog proračuna. Rezultati se veoma dobro slažu u srednjim katovima, na vrhu zgrade približna metoda daje nešto nepovoljnije rezultate, ali u prihvatljivim granicama.Authors propose an approximate method for calculating maximum horizontal displacement of unsymmetrical multistory buildings exposed to earthquake action. The method is known as the method of equivalent flexural stiffness. The multistory building of low, medium and high eccentricity is monitored. The obtained data are compared with appropriate results of nonlinear dynamic analysis. A very good correspondence has been registered for stories in the middle of the building, while results are somewhat less favorable for the top of the buildings, although they are still within acceptable limits

Defiences of design models for engeenering structures

Razmotreni su osnovni tipovi pogrešaka koje prate proračun konstrukcija. Upozorilo se I na one pogreške koje bi se znanjem I iskustvom mogle izbjegnuti. Posebno je upozoreno na pogreške koje nastaju pri tvorbi numeričkih modela jer se u tome dijelu kompjutorskih proračuna u praksi najviše griješi. Naglašene su I neizbježne pogreške, posebno one koje nastaju pri rješavanju velikih sustava jednadžbi. Pokušalo se I odgovoriti na pitanja kolika je pouzdanost rezultata naših proračuna.Basic types of errors occurring in the design of structures are considered. Particular attention is drawn to errors that could be avoided given appropriate knowledge and experience. A special emphasis is placed on errors occuring during creation of numerical models as it is precisely in in this part of computer design that most practical errors are made. Unavoidable errors are also presented, particulary those that occur while great equation systems are being solved. An attempt is made to provide answer to the question of the extent to which our calculation results can be considered reliable

Iterated Ritz Method for solving systems of linear algebraic equations

U radu je opisano stanje razvoja nove iteracijske metode za rješavanje sustava linearnih algebarskih jednadžbi. Metoda je pogodna za izrazito velike sustave slabo popunjenih matrica. Osim vlastitih obilježja, posjeduje i svojstvo općenitosti, jer su mnogi iteracijski postupci samo poseban slučaj ovoga pristupa. Algoritam je realiziran samostalno, a potom je pridružen programu otvorena koda FEAP. Provedene su i raznolike provjere, posebice na praktičnim modelima. Premda je postupak tek djelomice istražen, već pokazuje dobre rezultate.The paper describes research state of a new iterative method for solving systems of linear algebraic equations. The method is suitable for extremely large systems with sparse matrices. In addition to its own characteristics, it also has a feature of generality, as many iterative methods are only special cases of this approach. The algorithm was developed independently, and then implemented into the open source code program FEAP. Also, various checks were conducted, especially on practical models. Although the method has been only partially studied, good results have already been obtained

Break-even model application for business decision--making in olive production

Model točke pokrića je koristan analitički instrument za utvrđivanje ekonomskih učinaka koji pomaže u planiranju i donošenju poslovnih odluka u procesu upravljanja maslinarskom proizvodnjom. Ovaj model se koristi kao analitički instrument koji daje određene smjernice za kratkoročno poslovno odlučivanje. Cilj rada bio je primijeniti cjelokupnu analizu točke pokrića i pokazati odnose između troškova, prihoda i financijskog rezultata pri različitim razinama proizvodnje maslina te utvrditi razinu proizvodnje kod koje su pokriveni ukupni troškovi uz determiniranu prodajnu cijenu. Na temelju tehnoloških i ekonomskih polazišta izrađena je modelna kalkulacija pokrića varijabilnih troškova maslinarske proizvodnje. Modelnom kalkulacijom utvrđeni troškovi i prihodi predstavljali su podlogu za analizu točke pokrića i izračunavanje relevantnih ekonomskih veličina i pokazatelja. Pri količini proizvodnje od 386 kg maslinovog ulja ostvaruje se ukupni prihod u iznosu od 17.391,59 kn koji pokriva ukupne troškove, što znači da je potrebno ostvariti 28,9% ukupne količine proizvodnje za izlazak iz zone gubitka. Pozitivan financijski rezultat i vrijednosti ekonomskih pokazatelja proizvodnosti rada, ekonomičnosti i rentabilnosti proizvodnje u analitičkom modelu pokazuju da je konvencionalna proizvodnja maslina u punoj rodnosti efikasna pod pretpostavkom postizanja očekivanog priroda ploda, randmana ulja i njegove uspješne prodaje.The break-even model is an useful analytical instrument for determination of economic effects, which facilitates planning and business decision-making in the process of olive production management. This model is used as an analytical instrument that provides specific guidance for short-term business decision-making. The paper aim was to apply the whole break-even analysis, show the relations between costs, revenues, and financial result at various olive production levels, and determine the production level where total costs are covered with the determined selling price. Gross margin calculation model of olive production has been created on the basis of technological and economic starting points. Costs and revenues determined by model calculation were the basis for the break-even analysis and the calculation of relevant economic values and indicators. At the production of 386 kg of olive oil, the total revenue of 17,391.59 HRK covers the total cost, which means that 28.9% total production should be realised so as to come out of loss. The positive financial result and the values of economic indicators of labour productivity, cost-effectiveness and profitability of production in the analytical model show that the conventional production of olives in full fertility is effective under the assumption of realisation of the expected yield, the average yield of olive oil and the successful sales