Mathematics in Civil Engineering (anglicky) - předměty programu

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The subject is focused on qualitative properties of mathematical models of heat and moisture transport in porous materials. The lectures are devoted to derivation of models of transport processes in multiphase systems and solutions of corresponding initial-boundary value problems. The main topics of the subject: Balance equations, mass balance equations, energy balance equations, balance equations in multi-phase systems, heat and mass transport in porous materials, constitutive equations, Darcy’s law, Fourier’s law, Fick’s law, state equations, hygro-thermal parameters in transport models. Mathematical formulation of the problem, initial and boundary conditions. The method of Rothe, Faedo-Galerkin method. Solutions of elliptic problems generated by the method of discretization in time, existence and convergence theorem for the abstract parabolic problem, applications on simplified models of heat transport and isothermal moisture flow in porous materials. Coupled heat and moisture transport in porous materials.

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The goal is to make students familiar with basic mathematical tools provided by the computer algebra system (CAS) Maple. Course participants will acquire basic skills in applying Maple to solving mathematical and engineering problems. Topics: Differences between CAS and numerical software (Matlab, for instance). Maple core and packages. Maple worksheet and document modes; interaction with the user – palettes, context menu, line commands. Help system. Basic terms and operations: variable, expression, function, procedure, symbolic manipulation with expressions and functions, differentiation, integration, loops, conditional execution, assumptions, etc. Plots and animations, customizing plots (color, text, font, etc.), multiple plots. Exporting. Advanced tools: solving ordinary differential equations, initial and boundary value, problems, linear algebra. Numerical calculation. Course participants are expected to present one case study per student motivated by their research topic.

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The aim is to acquaint students with the basic problems of numerical mathematics. Thematic areas are: Systems of linear equations. Direct and basic iterative methods. Solving nonlinear equations and their systems Eigenvalue problem Approximation of functions Numerical quadrature Numerical methods of solving ordinary differential equations with initial and boundary conditions.

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The subject follows the Applied Mathematics and Numerical Methods I, the aim is to master methods of solving partial differential equations. Both elliptical and parabolic tasks will be solved. Less attention will be paid to hyperbolic problems. Problems of effective preconditioning of emerging systems of linear systems will also be addressed.

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The course covers selected chapters from the qualitative theory of ordinary differential equations and also of dynamical systems. Contents: Linear systems, nonlinear systems – local theory, nonlinear systems – global theory, nonlinear systems - selected topics of bifucation theory.

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The course covers selected topics of the theory of dynamical systems. Contents: Systems with stable asymptotic behavior, linear maps and linear differential equations, recurrence and equidistribution, conservative systems, simple systems with complicated orbit structure, entropy and chaos, hyperbolic dynamics, homoclinic tangles, strange attractors.

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Physical derivation of typical nonequilibrium problems in continuum physics, formulation and interpretation of initial and boundary conditions, classification of PDEs (parabolic, hyperbolic), solution methods (Galerkin, time discretization) including a survey of the basic theory of Sobolev spaces and embedding theorems, qualitative properties of solutions as, e. g., stability or instability of solution trajectories, occurrence of shock waves, or systems with memory.

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The lectures will be devoted to the study of Hilbert and Banach spaces and operators on them with regard to applications in the theory of partial differential equations. We say basic theorems of the functional analysis, Hahn-Banach's, Banach-Steinhaus's theorem, and the theorem on open mapping and on the closed graph. The concept of dual space and reflexivity, the quadratic functional, the theorem about the minimum and the relation with the operator equation have been introduced. Furthermore, we can prove Riesz's theorem on representation and Lax-Milgram's theorem. We will introduce a weak convergence and we will prove a weak compactness of the unit ball. We show that the convex continuous coercive functional in the reflexive Banach space has a minimum. Let's mention Browder's theorem about monotone operators. Finally, we will show applications on elliptical problems.

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Metric and topology of Euclidean spaces. Examples of fractal sets: Cantor set, Sierpinski triangke and carpet, Koch curve, Menger sponge. Elementary measure theory. Lebesgue measure, Hausdorff measure. Box-counting and Hausdorff dimension. Calculation of dimension. Examples. Self-similar sets. Iterated function systems. Hutchison operator. Attractors. Collation Theorem. Barnsley fern and other self-affine sets.

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Geostatistics is concerned with the estimation and prediction problems for stochastic phenomena on the Earth, using data obtained at a limited number of spatial locations called geodata. It refers to the application of general statistical principles of modeling and inference to geostatistical problems.

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The aim of this course is to provide doctoral students with an introduction in the theory of elliptic partial differential equations. The subjects of study are the following: the Laplace and Poisson equations, classical formulation of a boundary condition for the Laplace and Poisson equations, the Dirichlet, Neumann and Newton boundary conditions. Qualitative properties of solutions to the Laplace’s and Poisson’s equations, maximum principle, the Harnack inequality. A priory estimates of solutions and behavior of solutions near the boundary. Generalization of the qualitative theory of solutions to Laplace’s and Poisson’s equations for linear elliptic second order equations.

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Topology and metric in the plane and Euclidean spaces; convergence, continuous functions and mappings. Metric spaces. Topology of metric spaces, convergence, continuous functions and mappings, Urysohn Lemma, Tietze Theorem. Complete metric spaces, Banach Fixed Point Lemma. Compact metric spaces. Compactness in Euclidean spaces. Lipschitz and Holder functions. Topology on a set. Open and closed sets, closure, boundary. Urysohn Lemma, Tietze Theorem. Cartesian products, projections. Connected and totally disconnected spaces. Compactness. Tychonoff Theorem for finitely many spaces. Arzelá-Acoli Theorem. Stone-Weierstrass Theorem.

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Hilbert's spaces Bilinear forms and functionals Quadratic functional, symmetry, positive definitness, theorem about the minimum and relation to the equation Riesz's theorem and Lax-Milgram's theorem Finite element method, convergence (generally for nonsymmetric operator) - Riesz‘s and Galerkin‘s method It can converge slowly Better regularity converges better The least square method Variational crimes Selection of base functions: h-version, p-version, hp-version, hierarchical base, cascade Linear system preparation Methods of solution of the resulting systems - direct procedures - iterative procedures - possibilities of preconditioning

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The goal is to make students familiar with common methods for the minimization of functions of one or several real variables. Unconstrained as well as constrained minimization are considered. By using software tools (Matlab, SciLab, Octave, Python, etc.), course participants are expected to present a solution to a minimization problem motivated by the subject of their research. Topics: Minimization of functions of one real variable. Unconstrained minimization of functions of several real variables. Conditions for local optimality. Conjugate gradient method, quasi-Newton methods. Constrained minimization of functions of several real variables. Lagrange multipliers. Conditions for local optimality. Penalty method, active set method, gradient projection method, SQP method (Sequential Quadratic Programming). Introduction to linear programming, simplex method.

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The aim of the subject is to derive mathematical models of steady and nonsteady flow of incompressible fluids. Course contents: Vector and tenzor calculus, function spaces (Lebesque and Sobolev spaces), some known theorems of integral calculus that will be applied to derive mathematical models (Green's theorem, Stokes theorem, Gauss-Ostrograph's theorem), continuum and its kinematics, tenzor of small deformations, tenzor of velocity of deformation, Eulerian and Lagrangian description of motion, Reynolds transport theorem, the volume forces, the surface forces, the stress tenzor and its properties, constitutive equations, Stokesian fluids, basic types of Stokesian fluids: ideal fluid, Newtonian fluid, the pressure, the dynamic stress tensor, mathematical models of flow of incompressible fluid, formulation of boundary value problems for steady and nonsteady flow of incompressible fluid.

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Notion of time series. Stationary time series. Basic characteristics and their estimates. ARMA models. Frequency analysis of time series. Markovian sequences with finite number of states. Stationary distribution and method MCMC. Idea of MCMC for a continuous set of states.

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Distributions connected to normal distribution (chi square, t distribution). Multiple normal distribution and estimates of its parameters. Theory of estimation – a method of moments, a maximum likelihood method. Bayesian estimates. Method of principle components. Multiple linear regression. Non-linear regression. Bayesian approach to linear ans nonlinear regression.

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The goal of the subject is to inform students about the basics the mathematical theory of the Navier-Stokes equations for the incompressible fluid. The content of the subject: The description of the Navier-Stokes equations, the introduction of the fundamental concepts, the definition of the fundamental function spaces, the description of the basic relations between the function spaces, the definition of the classical and weak solution, the expulsion of the pressure from the definition of the weak solution, Helmholtz decomposition, some elementary properties of the weak solution, the proof of the existence of the weak solutions by the Galerkin method in a general domain, the discussion of several different definitions of the weak solution, qualitative properties of the weak solution, energy inequality, strong energy inequality, the sufficient conditions for the energy equality, the problem of the uniqueness and regularity, the fundamental uniqueness theorem, the role of the initial conditions, brief discussion of the asymptotic behavior of the solution, brief discussion of large solutions, brief discussion of various proofs of the existence of the weak solution, mild solutions.

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Students are introduced to basic computational methods related to the problems of linear algebra which can be obtained in engineering problems. The following topics are studied. Basics of linear algebra: vectors, matrices, systems of linear equations, solvability. Vector and matrix norms, eigenvalues and eigenvectors. Spectra of matrices. Coordinates with respect to a basis; change of a basis. Schur complement. Symmetric and positive definite matrices. Gauss elimination. LU decomposition. Matrix iterative methods: Jacobi method, Gauss-Seidel method. Gradient methods: method of steepest descent, conjugate gradient method. Convergence criteria and convergence rate. Conditioning of a system of linear equation. Preconditioning methods. Incomplete LU decomposition. Eigenproblems. Gram-Schmidt orthogonalization. Discrete Fourier transformation and its properties. Circulent matrix.

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The goal is to make students familiar with basic mathematical and numerical tools provided by Matlab, a language and environment for mostly numerical calculations. Course participants will acquire basic skills in applying Maple to solving mathematical and engineering problems. Topics: Matlab interactive environment, Matlab toolboxes. Basics of the Matlab language; vector, matrix, structure, variable, function; loop, conditional execution, m-file. Numerical iterative algorithms for solving nonlinear equations and systems of linear equations. Approximation and interpolation. The least-squares method, minimization of functions of several variables. Numerical integration. Plots and animations. Symbolic operations. Partial Differential Equation Toolbox, Optimization Toolbox, Global Optimization Toolbox. Course participants are expected to present one case study per student motivated by their research topic.

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The goal is to make students familiar with some non-stochastic methods for uncertainty quantification. Uncertainty is considered in parameters entering mathematical models. Consequently, the model output represented by a quantity of interest is also uncertain and this uncertainty is to be assessed. Topics: Aleatoric and epistemic uncertainty. Differential equations with uncertain data. Various approaches to uncertainty quantification. The worst- and best-case scenario method. Elements of fuzzy set theory (membership function, alpha-cut, Zadeh’s extension principle). Fuzzification, various definitions of membership functions, a connection to information gap theory by Y. Ben-Haim. An introduction to the Dempster-Shafer theory (DST), belief and plauzibility, Dempster’s rule of combination. Probabilistic interpretation of DST. Application to engineering problems with uncertain data and a non-trivial state problem. Tools for solving such problems – minimization algorithms, sensitivity analysis, finite element method.

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Basic principles of object-oriented programming (C ++, D, ADA, Fortran), algorithms design, component programming, coexistence of different platforms, portability of programs on various hardware platforms, security aspects of programming.

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This course provides an introduction to the mathematical theory of ordinary differential equations and methods of finding their solutions. Many models in engeneering can be expressed as differential equations. Knowledge how to select and use an apropriate model and techniques for finding its solutions is essential for scientists and engineers. The course covers core topics such as first order differential equations (separable equations, exact equations, homogeneous equations, linear equations, the Bernoulli and Riсcati equations), initial and boundary value problems, linear higher-order differential equations, systems of differential equations.

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The lectures will be devoted to the study of function spaces with respect to applications in the theory of partial differential equations. The Lebesgue and Sobolev spaces will be introduced. The teorem about the density of smooth functions and the theorem about extending the operator from a dense subset will be proved. In addition, the Hausdorff measure will be introduced, and Lebesgue spaces on the boundary and non-integer order spaces will be defined. Proofs of embeddings, trace, inverse traces, and compact embeddings will be proved. Finally, we will show applications on elliptic problems.

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The course covers selected topics of the theory of nonnegative matrices and also of positive operators. Contents: Matrices that leave a cone invariant, nonnegative matrices, semigroups of nonnegative matrices, iterative methods for linear systems, finite Markov chains, examples. Positive operators, spectral theory of positive operators, examples and possible applications.

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The goal of the subject is to inform students about the basics of the regularity theory for the weak solutions of the Navier-Stokes equtions (NSE) for the incompressible fluid. The content of the subject: the description of NSE, the introduction of the fundamental concepts from the mathematical theory of NSE, the definition of the basic function spaces, the definition of the weak solution, a brief proof of the existence of the weak solution by the Galerkin method, structure theorem, epochs of irregularity, Hausdorff measure and dimension, parabolic measure, the size of the set of time singular points, the definition of the suitable solution, regular and singular points in spacetime, partial regularity, local regularity conditions, dimension of the set of singular points, conditional regularity, Prodi-Serrin conditions, conditional regularity in terms of one or two components of the velocity field, conditional regularity in terms of some items of the velocity gradient, conditional regularity in terms of pressure, pressure gradient, vorticity and other quantities.

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Robust statistics deals with statistical methods which are not sensitive to small depatrures from model assumptions. Robust statistics became a part of mainstream statistics only a few years ago yet it is implemented in many statistical softwars.

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Students are introduced to basic solution methods for problems dependent on random input variables and for estimating of models and their parameters from measured data. The subject focuses on the computational properties of these methods, related numerical methods, their convergence conditions and efficiency. The particular topics are numerical solution of determinic partial differential equations, finite element method, finite difference method (only a sketch of them both); basic methods of computational probability; partial differenetial equations with random parameters; Monte Carlo Method; collocation method; stochastic Gallerkin method; solution spaces of problems with random data; Karhunen-Loeve expansion; Mercer’s lemma; covariance matrix decomposition; convergence with respect to random variables; Bayesian methods; inverse analysis.

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The subject is focused on variational formulations and solutions of fundamental static and quasistatic problems in the mathematical theory of elasticity. The lectures are devoted to boundary value problems for elliptic equations with emphasis on the problems of the existence and uniqueness of solutions. The main topics of the subject: Stress tensor, equations of equilibrium, strain tensor, equations of the compatibility of strain, Hooke’s law, the spaces of functions with finite energy, classical and variational formulations of boundary value problems in the theory of elasticity, Rellich's theorem, coerciveness of strains, Korn's inequality, coercive and weakly lower semi-continuous functionals, differentiability in the Gateaux sense, solvability of problems in the theory of elasticity, variational principle, elasto-inelastic bodies, models with internal state variables

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Iterative methods of solving systems of linear algebraic equations. Fast algorithms. Gradient methods. CG and GMRES. Preconditioning and its methods. Multilevel Methods for Elliptical Problems (Multigrid Methods). V-cycle, W-cycle. Domain Decomposition Methods (DDM). Overlapping methods, methods without overlapping. Neumann-Neumann methods. Balanced DD method.Methods of Scwarz type (Fully Black Box Methods). Special methods for non-eliptic and indefinite problems. Typical tasks, Helmholtz equation, Navier-Stokes system. Aggregation of the Leontief System. Stationary vectors of probability of stochastic matrices. All methods and algorithms are interconnected and illustrated on non-academic examples of models of mechanics, elasticity, strength and reliability of buildings.

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The course will cover analytical methods for multiscale modeling of heterogenous materials, with emphasis on: 1. Introduction, overview of governing equations of elasticity, tensor notation, and averaging 2. Minimum energy principles, material symmetries 3. Elementary theory of overall moduli, concentration factors, Voigt-Reuss bounds 4. Exact solution for two-phase composites, idea of improved bounds 5. Eshelby problem 6. Approximate evaluation of overall moduli: dilute approximation, self-consistent method, Mori-Tanaka method 7. Improved bounds on overall moduli: Hashin-Shtrikman bounds 8. Thermo-elasticity 9. Extension to stationary transport processes

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The course will cover numerical methods for multiscale modeling of heterogenous materials, with emphasis on: 1. Overview of the finite element method for elasticity and heat conduction 2. Introduction to the method of asymptotic expansion for heat conduction and elasticity 3. First-order computational homogenization for elasticity 4. First-order computational homogenization for heat conduction and thermo-elasticity 5. Homogenization nonlinear problems -- application to non-linear conduction and elasticity 6. Two-scale simulations -- basic principles and implementation strategy, applications Reduced-order models, combining computational homogenization with micromechanics

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The course focuses at systematic mathematical description of mechanical behavior of materials. The subjects include: The model of continuum and the concept of representative volume element. General principles of constitutive modeling. Theories of elasticity (hyperelasticity, Cauchy elasticity, hypoelasticity). Viscoelasticity and the theory of creep. Yield and failure criteria. Incremental theory of plasticity. Damage mechanics. Fracture mechanics. Fatigue.

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This course covers the fundamentals of tensor algebra and calculus and demonstrates the power of tensor notation applied to formulation and solution of engineering problems. Selected examples cover solid and fluid mechanics, as well as heat and mass transport problems. The first part of the course is devoted to the definition of tensors, understood as linear mappings, to algebraic operations with tensors, to tensor fields and their differentiation, and to transformations between volume and surface integrals based on the Green and Gauss theorems. In the second part, it is shown how these mathematical tools enable an elegant description and analysis of various physical problems, with focus on applications in civil and mechanical engineering. The classes combine lectures and seminars, with emphasis on problems assigned as homework, which form the basis of presentations and discussions in class. The objective is not only to transfer specific knowledge, but also to develop the students‘ aptitude for independent thinking and critical analysis. At the same time, mastering of tensorial notation by the students will greatly facilitate their future reading of modern scientific literature in many fields of research.

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Critical Zone is defined as a thin layer of the Earth’s surface and near-surface terrestrial environment from the top of the vegetation canopy, or atmosphere–vegetation interface, to the bottom of the weathering zone, or freshwater–bedrock interface (US National Research Council, 2001). A variety of physical, chemical and biological interactions between the biotic and abiotic constituents of the critical zone occurs over a range of spatial and temporal scales. These interactions determine near surface fluxes of mass, energy and momentum and control transport and cycling of water, carbon and other chemicals. Understanding critical zone processes is an important prerequisite for the prediction of the consequences of surface pollution, climate change impacts and land use adaptation effects. The course aims at making students understand basic principles facilitating the quantitative description of the state and flow of water and transport of dissolved chemicals and energy in the critical zone, with emphasis on the processes crucial for the soil–plant–atmosphere system. The course covers the topics of parameterization of soil and plant hydraulic properties; formulation of governing equations of water flow, solute transport and heat transfer; initial and boundary conditions of the governing equations and basic measurement techniques. Specific attention will be paid to the individual hydraulic and transport processes, such as: infiltration, evaporation, redistribution, capillary rise, plant root water uptake, sap flow and plant transpiration, surface and subsurface stormflow, preferential flow and transport of contaminants in the soil profile.

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The course focuses on processes affecting the transport of water-dissolved chemicals in natural porous systems. In addition to reactive substances, conceptualization of basic transport processes is considered for non-reactive (conservative) substances. The emphasis is put on mathematical description of water movement and transport of solutes in soils including transformation processes (initial and boundary conditions, governing equations, analytical and numerical solution, parameterization of transport coefficients). Attention is also be paid to the concept of breakthrough curve, parameterization of sorption and degradation processes, description of chemical transport using a two-region/two-site model, existing databases of transport parameters of organic compounds, and different approaches used to assess mobility of organic compounds in the vadose zone.

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The course aims to provide advanced knowledge of two- and three-phase flows (a combination of phases: liquid – solids (particles) gas) with applications in pressurized pipes and open channels. Fundamental principles are discussed of flow of mixture with particular attention paid to mechanisms governing behaviour of mixture flow (dispersion, sedimentation, boundary friction and inner friction, including effects of Newtonian and non-Newtonian carrier). Theories and on theories based predictive models are introduced and their application demonstrated on practical case studies of e.g. pumping and transport of sludge in technological processes, hydraulic transport of solids in pipelines and launders, or sediment transport in rivers and streams. Also discussed are examples of computation of multiphase flows in commercial software including CFD (Computational Fluid Dynamics) software. In the course, each student submits his/her seminar work on a chosen subject.