Computational Fluid Dynamics (CFD) has been widely used as a tool to analyze physical phenomena involving fluids. To perform a CFD simulation, the governing equations are discretized to formulate a set of nonlinear algebraic equations. Typical spatial discretization schemes include finite-difference methods, finite-volume methods, and finite-element methods. Error introduced in the discretization process is called discretization error and defined as the difference between the exact solution to the discrete equations and the exact solution to the partial differential or integral equations. For most CFD simulations, discretization error accounts for the largest portion of the numerical error in the simulation. Discretization error has a complicated nonlinear relationship with the computational grid and the discretization scheme, which makes it especially difficult to estimate. Thus, it is important to study the discretization error to characterize numerical errors in a CFD simulation. Discretization error estimation is performed using the Error Transport Equations (ETE) in this work. The original nonlinear form of the ETE can be linearized to formulate the linearized ETE. Results of the two types of the ETE are compared. This work implements the ETE for finite-volume methods and Discontinuous Galerkin (DG) finite-element methods. For finite volume methods, discretization error estimates are obtained for both steady state problems and unsteady problems. The work on steady-state problems focuses on turbulent flow modelled by the Spalart-Allmaras (SA) model and Menter's $k-omega$ SST model. Higher-order discretization error estimates are obtained for both the mean variables and the turbulence working variables. The type of pseudo temporal discretization used for the steady-state problems does not have too much influence on the final converged solution. However, the temporal discretization scheme makes a significant difference for unsteady problems. Different temporal discretizations also impact the ETE implementation. This work discusses the implementation of the ETE for the 2-step Backward Difference Formula (BDF2) and the Singly Diagonally Implicit Runge-Kutta (SDIRK) methods. Most existing work on the ETE focuses on finite-volume methods. This work also extends ETE to work with the DG methods and tests the implementation with steady state inviscid test cases. The discretization error estimates for smooth test cases achieve the expected order of accuracy. When the test case is non-smooth, the truncation error estimation scheme fails to generate an accurate truncation error estimate. This causes a reduction of the discretization error estimate to first-order accuracy. Discussions are made on how accurate truncation error estimates can be found for non-smooth test cases. / Doctor of Philosophy / For a general practical fluid flow problem, the governing equations can not be solved analytically. Computational Fluid Dynamics (CFD) approximates the governing equations by a set of algebraic equations that can be solved directly by the computer. Compared to experiments, CFD has certain advantages. The cost for running a CFD simulation is typically much lower than performing an experiment. Changing the conditions and geometry is usually easier for a CFD simulation than for an experiment. A CFD simulation can obtain information of the entire flow field for all field variables, which is nearly impossible for a single experiment setup. However, numerical errors are inherently persistent in CFD simulations due to the approximations made in CFD and finite precision arithmetic of the computer. Without proper characterization of errors, the accuracy of the CFD simulation can not be guaranteed. Numerical errors can even result in false flow features in the CFD solution. Thus, numerical errors need to be carefully studied so that the CFD simulation can provide useful information for the chosen application. The focus of this work is on numerical error estimation for the finite-volume method and the Discontinuous Galerkin (DG) finite-element method. In general, discretization error makes the most significant contribution to the numerical error of a CFD simulation. This work estimates discretization error by solving a set of auxiliary equations derived for the discretization error of a CFD solution. Accurate discretization error estimates are obtained for different test cases. The work on the finite-volume method focus on discretization error estimation for steady state turbulent test cases and unsteady test cases. To the best of the author's knowledge, the implementation of the current discretization error estimation scheme has only been applied as an intermediate step for the error estimation of functionals for the DG method in the literature. Results for steady-state inviscid test cases for the DG method are presented.

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Computational fluid dynamics (CFD) has proven to be an invaluable tool for both engineering design and analysis. As the performance of engineering devices become more reliant upon the accuracy of CFD simulations, it is necessary to not only quantify and but also to reduce the numerical error present in a solution. Discretization error is often the primary source of numerical error. Discretization error is introduced locally into the solution by truncation error. Truncation error represents the higher order terms in an infinite series which are truncated during the discretization of the continuous governing equations of a model. Discretization error can be reduced through uniform grid refinement but is often impractical for typical engineering problems. Grid adaptation provides an efficient means for improving solution accuracy without the exponential increase in computational time associated with uniform grid refinement. Solution accuracy can be improved through local grid refinement, often referred to as h-adaptation, or by node relocation in the computational domain, often referred to as r-adaptation. The goal of this work is to examine the effectiveness of several r-adaptation techniques for reducing discretization error. A framework for geometry preservation is presented, and truncation error is used to drive adaptation. Sample problems include both subsonic and supersonic inviscid flows. Discretization error reductions of up to an order of magnitude are achieved on adapted grids. / Master of Science

Os custos econômicos e perdas potenciais de vidas humanas resultantes de falhas em sistemas de drenagem de águas pluviais podem ser enormes, aumentando a necessidade da busca de melhores métodos de dimensionamento destes. Os métodos baseados na precipitação de projeto são os mais usados para a estimativa do hidrograma de projeto em bacias urbanas, devido às facilidades que os mesmos apresentam em relação aos outros métodos. Entretanto, estes métodos têm suas limitações e incertezas, que podem influenciar as características do hidrograma de projeto resultante, incertezas que são, em consequência, transferidas ao dimensionamento hidráulico das obras propostas. No presente trabalho, foram avaliados os efeitos da discretização espacial da bacia, da discretização temporal e da posição do pico do hietograma de projeto, sobre as principais características do hidrograma de projeto. Esses efeitos foram ainda avaliados no dimensionamento hidráulico do sistema de macrodrenagem dos bairros de Mavalane “A” e Maxaquene “A”, localizados na cidade de Maputo em Moçambique. Para avaliar os referidos efeitos, a bacia foi discretizada em 1, 5, 7 e 12 sub-bacias. O intervalo de tempo do hietograma de projeto foi discretizado em 1, 2, 3, 4 e 5 minutos, sendo calculados para cada discretização e intervalo de tempo, hietogramas com pico no início, 25%, 50%, 75% e final do evento com duração correspondente a 90 minutos e 24 horas. Os hietogramas foram transformados em vazão no programa IPHS1, a qual foi propagada no sistema de macrodrenagem usando o modelo EPA SWMM 5.0, sendo obtidos os hidrogramas de projeto resultantes. O exutório da bacia e o trecho C19 localizado na região central da bacia foram escolhidos para análise dos resultados. Os resultados obtidos mostram uma influência importante dos efeitos avaliados nas características do hidrograma de projeto: vazão de pico, tempo de pico e volume parcial e, consequentemente, no dimensionamento da rede de macrodrenagem, sendo o efeito da discretização da bacia o mais influente, seguido dos efeitos da posição do pico da chuva e intervalo de tempo. / The economic costs and potential losses of human lives due to failure in stormwater drainage systems can be significant, calling for an improvement in methods used for its design. The design storm approach, also known as “single-event design-storm” is widespread used in formulating design hydrograph from historical rainfall data in urban watersheds, due to its relative advantages when compared to other methods. Therefore, this approach has its uncertainties which can affect the resulted design hydrograph, and consequently affect the hydraulic design of proposed structures. In this study is presented an assessment of the adopted values in design criteria, during the estimation of design hydrograph used in hydraulic design of stormwater drainage systems. In particular, are assessed the effects of spatial discretization of the watershed, the temporal discretization and peak position of the design hyetograph, in the main features of the resulted design hydrograph. The same effects are then assessed in the hydraulic design of the major stormwater drainage system of Mavalane “A” and Maxaquene “A” neighborhoods, located in Maputo city in Mozambique. To assess these effects, the watershed was discretized into 1, 5, 7 and 12 sub-watersheds. The time step of the design hyetograph was discretized into 1, 2, 3, 4 and 5 minutes, and then computed for each discretization and time step, hyetographs with peak positioned in the beginning, 25%, 50%, 75% and the end of the duration of 90 minutes and 24 hours. The hyetographs were converted into runoff in IPHS1, wish was dynamically routed in the drainage system, using EPA SWMM 5.0. The watershed outlet and conduit C19 were used to analyze the results. The results show an important influence of the assessed effects on the design hydrograph features: hydrograph peak, time to peak and partial volume and, consequently, on the hydraulic design of the major stormwater drainage system, being the effect of spatial discretization of the watershed, the most important, followed by hyetograph peak position and time step.

Os custos econômicos e perdas potenciais de vidas humanas resultantes de falhas em sistemas de drenagem de águas pluviais podem ser enormes, aumentando a necessidade da busca de melhores métodos de dimensionamento destes. Os métodos baseados na precipitação de projeto são os mais usados para a estimativa do hidrograma de projeto em bacias urbanas, devido às facilidades que os mesmos apresentam em relação aos outros métodos. Entretanto, estes métodos têm suas limitações e incertezas, que podem influenciar as características do hidrograma de projeto resultante, incertezas que são, em consequência, transferidas ao dimensionamento hidráulico das obras propostas. No presente trabalho, foram avaliados os efeitos da discretização espacial da bacia, da discretização temporal e da posição do pico do hietograma de projeto, sobre as principais características do hidrograma de projeto. Esses efeitos foram ainda avaliados no dimensionamento hidráulico do sistema de macrodrenagem dos bairros de Mavalane “A” e Maxaquene “A”, localizados na cidade de Maputo em Moçambique. Para avaliar os referidos efeitos, a bacia foi discretizada em 1, 5, 7 e 12 sub-bacias. O intervalo de tempo do hietograma de projeto foi discretizado em 1, 2, 3, 4 e 5 minutos, sendo calculados para cada discretização e intervalo de tempo, hietogramas com pico no início, 25%, 50%, 75% e final do evento com duração correspondente a 90 minutos e 24 horas. Os hietogramas foram transformados em vazão no programa IPHS1, a qual foi propagada no sistema de macrodrenagem usando o modelo EPA SWMM 5.0, sendo obtidos os hidrogramas de projeto resultantes. O exutório da bacia e o trecho C19 localizado na região central da bacia foram escolhidos para análise dos resultados. Os resultados obtidos mostram uma influência importante dos efeitos avaliados nas características do hidrograma de projeto: vazão de pico, tempo de pico e volume parcial e, consequentemente, no dimensionamento da rede de macrodrenagem, sendo o efeito da discretização da bacia o mais influente, seguido dos efeitos da posição do pico da chuva e intervalo de tempo. / The economic costs and potential losses of human lives due to failure in stormwater drainage systems can be significant, calling for an improvement in methods used for its design. The design storm approach, also known as “single-event design-storm” is widespread used in formulating design hydrograph from historical rainfall data in urban watersheds, due to its relative advantages when compared to other methods. Therefore, this approach has its uncertainties which can affect the resulted design hydrograph, and consequently affect the hydraulic design of proposed structures. In this study is presented an assessment of the adopted values in design criteria, during the estimation of design hydrograph used in hydraulic design of stormwater drainage systems. In particular, are assessed the effects of spatial discretization of the watershed, the temporal discretization and peak position of the design hyetograph, in the main features of the resulted design hydrograph. The same effects are then assessed in the hydraulic design of the major stormwater drainage system of Mavalane “A” and Maxaquene “A” neighborhoods, located in Maputo city in Mozambique. To assess these effects, the watershed was discretized into 1, 5, 7 and 12 sub-watersheds. The time step of the design hyetograph was discretized into 1, 2, 3, 4 and 5 minutes, and then computed for each discretization and time step, hyetographs with peak positioned in the beginning, 25%, 50%, 75% and the end of the duration of 90 minutes and 24 hours. The hyetographs were converted into runoff in IPHS1, wish was dynamically routed in the drainage system, using EPA SWMM 5.0. The watershed outlet and conduit C19 were used to analyze the results. The results show an important influence of the assessed effects on the design hydrograph features: hydrograph peak, time to peak and partial volume and, consequently, on the hydraulic design of the major stormwater drainage system, being the effect of spatial discretization of the watershed, the most important, followed by hyetograph peak position and time step.

Liquid jet atomisation has a wide variety of application in areas such as injectors in automobile and launch vehicle combustors, spray painting, ink jet printing etc. Understanding physical mechanisms involved in the primary regime of atomisation in combustors is extremely challenging due to the lack of experimental techniques that can reliably provide measurements of gas and liquid velocity fields in this region. Experimental studies have so far been mostly restricted to conditions at atmospheric conditions rather than technically relevant operating pressures. We present a computational fluid dynamics based modelling approach that can capture the evolution of the flow field in the dense primary atomization region of the spray as part of the present thesis work. A fully compressible 3D flow solver is coupled with an interface tracking solver based on level set method. A generalised mathematical formulation for thermodynamic models is implemented in flow solver enabling easy switching between various equations of states. Solvers are parallelised to run on large number of processors and are shown to have good scalability. A modification to the level set method which greatly reduces mass conservation inaccuracies when compared with existing state-of-art baseline schemes has been developed during this work. The Ghost uid Method is used for applying matching conditions at the Interface. The liquid and gas phases are modelled using the perfect gas and Tait equations of state respectively. Several validation studies have been carried out to ensure quantitative accuracy of the solver implemented. Results from canonical Rayleigh Taylor instability simulations shows good agreement with reported results in literature. Finally, results for unsteady evolution of a water-air jet at a liquid to gas density ratio of 10 are shown. Physical mechanisms causing the initial droplet formation are discussed in detail. Droplet feedback is identified as one of the important mechanisms in triggering liquid core instabilities. Comparisons between droplet size distributions obtained from computations are carried out. Vorticity dynamics is used to understand hole and ligament formation from liquid core. Effect of numerical droplets on the simulation results is also looked at in detail.

Os custos econômicos e perdas potenciais de vidas humanas resultantes de falhas em sistemas de drenagem de águas pluviais podem ser enormes, aumentando a necessidade da busca de melhores métodos de dimensionamento destes. Os métodos baseados na precipitação de projeto são os mais usados para a estimativa do hidrograma de projeto em bacias urbanas, devido às facilidades que os mesmos apresentam em relação aos outros métodos. Entretanto, estes métodos têm suas limitações e incertezas, que podem influenciar as características do hidrograma de projeto resultante, incertezas que são, em consequência, transferidas ao dimensionamento hidráulico das obras propostas. No presente trabalho, foram avaliados os efeitos da discretização espacial da bacia, da discretização temporal e da posição do pico do hietograma de projeto, sobre as principais características do hidrograma de projeto. Esses efeitos foram ainda avaliados no dimensionamento hidráulico do sistema de macrodrenagem dos bairros de Mavalane “A” e Maxaquene “A”, localizados na cidade de Maputo em Moçambique. Para avaliar os referidos efeitos, a bacia foi discretizada em 1, 5, 7 e 12 sub-bacias. O intervalo de tempo do hietograma de projeto foi discretizado em 1, 2, 3, 4 e 5 minutos, sendo calculados para cada discretização e intervalo de tempo, hietogramas com pico no início, 25%, 50%, 75% e final do evento com duração correspondente a 90 minutos e 24 horas. Os hietogramas foram transformados em vazão no programa IPHS1, a qual foi propagada no sistema de macrodrenagem usando o modelo EPA SWMM 5.0, sendo obtidos os hidrogramas de projeto resultantes. O exutório da bacia e o trecho C19 localizado na região central da bacia foram escolhidos para análise dos resultados. Os resultados obtidos mostram uma influência importante dos efeitos avaliados nas características do hidrograma de projeto: vazão de pico, tempo de pico e volume parcial e, consequentemente, no dimensionamento da rede de macrodrenagem, sendo o efeito da discretização da bacia o mais influente, seguido dos efeitos da posição do pico da chuva e intervalo de tempo. / The economic costs and potential losses of human lives due to failure in stormwater drainage systems can be significant, calling for an improvement in methods used for its design. The design storm approach, also known as “single-event design-storm” is widespread used in formulating design hydrograph from historical rainfall data in urban watersheds, due to its relative advantages when compared to other methods. Therefore, this approach has its uncertainties which can affect the resulted design hydrograph, and consequently affect the hydraulic design of proposed structures. In this study is presented an assessment of the adopted values in design criteria, during the estimation of design hydrograph used in hydraulic design of stormwater drainage systems. In particular, are assessed the effects of spatial discretization of the watershed, the temporal discretization and peak position of the design hyetograph, in the main features of the resulted design hydrograph. The same effects are then assessed in the hydraulic design of the major stormwater drainage system of Mavalane “A” and Maxaquene “A” neighborhoods, located in Maputo city in Mozambique. To assess these effects, the watershed was discretized into 1, 5, 7 and 12 sub-watersheds. The time step of the design hyetograph was discretized into 1, 2, 3, 4 and 5 minutes, and then computed for each discretization and time step, hyetographs with peak positioned in the beginning, 25%, 50%, 75% and the end of the duration of 90 minutes and 24 hours. The hyetographs were converted into runoff in IPHS1, wish was dynamically routed in the drainage system, using EPA SWMM 5.0. The watershed outlet and conduit C19 were used to analyze the results. The results show an important influence of the assessed effects on the design hydrograph features: hydrograph peak, time to peak and partial volume and, consequently, on the hydraulic design of the major stormwater drainage system, being the effect of spatial discretization of the watershed, the most important, followed by hyetograph peak position and time step.

Cette thèse présente une nouvelle classe de schémas de discrétisation spatiale sur maillages polyédriques, nommée Compatible Discrete Operator (CDO) et en étudie l'application aux équations elliptiques et de Stokes. La préservation au niveau discret des caractéristiques essentielles du système continu sert de fil conducteur à la construction des opérateurs. Les opérateurs de de Rham définissent les degrés de liberté en accord avec la nature physique des champs à discrétiser. Les équations sont décomposées de manière à différencier les relations topologiques (lois de conservation) des relations constitutives (lois de fermeture).Les relations topologiques sont associées à des opérateurs différentiels discrets et les relations constitutives à des opérateurs de Hodge discrets. Une particularité de l'approche CDO est l'utilisation explicite d'un second maillage, dit dual, pour bâtir l'opérateur de Hodge discret. Deux familles de schémas CDO sont ainsi considérées : les schémas vertex-based lorsque le potentiel est discrétisé aux sommets du maillage (primal), et les schémas cell-based lorsque le potentiel est discrétisé aux sommets du maillage dual (les sommets duaux étant en bijection avec les cellules primales).Les schémas CDO associés à ces deux familles sont présentés et leur convergence est analysée. Une première analyse s'appuie sur une définition algébrique de l'opérateur de Hodge discret et permet d'identifier trois propriétés clés : symétrie, stabilité et $mathbb{P}_0$-consistance. Une seconde analyse s'appuie sur une définition de l'opérateur de Hodge discret à l'aide d'opérateurs de reconstruction pour lesquels sont identifiées les propriétés à satisfaire. Par ailleurs, les schémas CDO fournissent une vision unifiée d'une large gamme de schémas de la littérature (éléments finis, volumes finis, schémas mimétiques…).Enfin, la validité et l'efficacité de l'approche CDO sont illustrées sur divers cas tests et plusieurs maillages polyédriques / This thesis presents a new class of spatial discretization schemes on polyhedral meshes, called Compatible Discrete Operator (CDO) schemes and their application to elliptic and Stokes equations. In CDO schemes, preserving the structural properties of the continuous equations is the leading principle to design the discrete operators. De Rham maps define the degrees of freedom according to the physical nature of fields to discretize. CDO schemes operate a clear separation between topological relations (balance equations) and constitutive relations (closure laws).Topological relations are related to discrete differential operators, and constitutive relations to discrete Hodge operators. A feature of CDO schemes is the explicit use of a second mesh, called dual mesh, to build the discrete Hodge operator. Two families of CDO schemes are considered: vertex-based schemes where the potential is located at (primal) mesh vertices, and cell-based schemes where the potential is located at dual mesh vertices (dual vertices being in one-to-one correspondence with primal cells).The CDO schemes related to these two families are presented and their convergence is analyzed. A first analysis hinges on an algebraic definition of the discrete Hodge operator and allows one to identify three key properties: symmetry, stability, and $mathbb{P}_0$-consistency. A second analysis hinges on a definition of the discrete Hodge operator using reconstruction operators, and the requirements on these reconstruction operators are identified. In addition, CDO schemes provide a unified vision on a broad class of schemes proposed in the literature (finite element, finite element, mimetic schemes...).Finally, the reliability and the efficiency of CDO schemes are assessed on various test cases and several polyhedral meshes

Dans quelle mesure peut-on lire les propriétés dynamiques (quand le temps tend vers l’infini) d’un système sur des simulations numériques ? Pour tenter de répondre à cette question, on étudie dans cette thèse un modèle rendant compte de ce qui se passe lorsqu’on calcule numériquement les orbites d’un système à temps discret f (par exemple un homéomorphisme). L’ordinateur travaillant à précision numérique finie, il va remplacer f par une discrétisation spatiale de f, notée f_N (où l’ordre de la discrétisation N rend compte de la précision numérique). On s’intéresse en particulier au comportement dynamique des applications finies f_N pour un système f générique et pour l’ordre N tendant vers l’infini, où générique sera à prendre dans le sens de Baire (principalement parmi des ensembles d’homéomorphismes ou de C^1-difféomorphismes). La première partie de cette thèse est consacrée à l’étude de la dynamique des discrétisations f_N lorsque f est un homéomorphisme conservatif/dissipatif générique d’une variété compacte. L’étude montre qu’il est illusoire de vouloir retrouver la dynamique du système de départ f à partir de celle d’une seule discrétisation f_N : la dynamique de f_N dépend fortement de l’ordre N. Pour détecter certaines dynamiques de f il faut considérer l’ensemble des discrétisations f_N, lorsque N parcourt N.La seconde partie traite du cas linéaire, qui joue un rôle important dans l’étude du cas des C^1-difféomorphismes génériques, abordée dans la troisième partie de cette thèse. Sous ces hypothèses, on obtient des résultats similaires à ceux établis dans la première partie, bien que plus faibles et de preuves plus difficiles. / How is it possible to read the dynamical properties (ie when the time goes to infinity) of a system on numerical simulations ? To try to answer this question, we study inthis thesis a model reflecting what happens when the orbits of a discrete time system f (for example an homeomorphism) are computed numerically. The computer working in finite numerical precision, it will replace f by a spacial discretization of f, denotedby f_N (where the order N of discretization stands for the numerical accuracy). In particular, we will be interested in the dynamical behaviour of the finite maps f_N for a generic system f and N going to infinity, where generic will be taken in the sense of Baire (mainly among sets of homeomorphisms or C^1-diffeomorphisms). The first part of this manuscript is devoted to the study of the dynamics of the discretizations f_N, when f is a generic conservative/dissipative homeomorphism of a compact manifold. We show that it would be mistaken to try to recover the dynamics of f from that of a single discretization f_N : its dynamics strongly depends on the order N. To detect some dynamical features of f we have to consider all thediscretizations f_N when N goes through N.The second part deals with the linear case, which plays an important role in the study of C^1-generic diffeomorphisms, discussed in the third part of this manuscript. Under these assumptions, we obtain results similar to those established in the first part,though weaker and harder to prove.

Baigiamajame magistro darbe nagrinėjami struktūrų modeliavimo, diskretizavimo ir optimizavimo uždaviniai, jų sprendimo būdai ir algoritmai. Pasiūlyta originali strypinių struktūrų optimizavimo ir diskretizavimo technika, kurioje naudojami vaizdų algebros metodai ir baigtinių elementų metodo (toliau – BEM) programiniai paketai. Atlikta BEM programinių paketų apžvalga, parinkti tinkamiausi paketai darbo tikslams realizuoti ir rezultatams verifikuoti. Išanalizuoti skaitmeninių vaizdų skeletų išgavimo metodai. Pasiūlytas originalus vaizdo skeleto apdorojimo algoritmas skerspjūvio charakteristikoms nustatyti. Sudarytas ir programiškai realizuotas strypinių struktūrų optimizavimo-diskretizavimo algoritmas. Realizuota programinė sistema susideda iš vaizdų apdorojimo ir duomenų paruošimo dalies (MATLAB kalba) bei BEM skaičiavimų dalies (ANSYS vidine programavimo kalba APDL). Skaičiavimo rezultatai atvaizduojami ir verifikuojami STAAD.Pro paketu. Diskretizavimo metu strypinės struktūros mazgų vieta randama vaizdo skeleto segmentų sankirtos taškuose, o segmentų skerspjūvių plotai randami atkarpų, jungiančių šiuos mazgus, vidurio taškuose. Kai struktūros mazgų padėtis fiksuota arba mazgai yra per arti vienas kito atliekamas mazgų padėties koregavimas. Darbe atlikti testiniai skaičiavimai, rezultatų analizė ir verifikavimas, suformuluotos išvados. / In the master thesis the problems of structure modeling, discretization-optimization and their solution methods and algorithms are analyzed. The original technique for optimization and discretization of beam structures has been suggested; The packages of image algebra methods and of the finite element methods were employed for that. Several packages of finite element method have been reviewed and the most suitable packages for the current problems were identified. The methods for obtaining skeletons of digital images were explored. The algorithms for optimization and discretization of beam structures has been suggested and coded. The program created consents of the part for image processing and input data preparing, and the part for image the finite element via method. The results obtained are represented and verified by STAAD.Pro package. During the discretization, the positions of structure nodes are obtained in the intersection of skeleton segments. The segments' cross-section areas are obtained in the middle-points between two adjacent nodes. The positions of nodes may be corrected if the nodes close to each other. The test-calculation, analysis of results and verification are presented and conclusions are drawn.