lecture 1 - CE 30125 - Lecture 1 LECTURE 1 INTRODUCTION...

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CE 30125 - Lecture 1 p. 1.1 LECTURE 1 INTRODUCTION Formulating a “Mathematical” Model versus a Physical Model • Formulate the fundamental conservation laws to mathematically describe what is physi- cally occurring. Also define the necessary constitutive relationships (relate variables based on observations) and boundary conditions (b.c.’s) and/or compatibility constraints. • Use the laws of physics applied to an object/domain to develop the governing equations. Algebraic equations Integral equations valid for the domain as a whole p.d.e.’s valid at every point within the domain • e.g. Newton’s 2nd law applied to a point in a hypothetical continuum Navier- Stokes equations • Solve the resulting equations using • Analytical solutions • Numerical or discrete solutions • Verify how well you have solved the problem by comparing to measurements
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CE 30125 - Lecture 1 p. 1.2 INSERT FIGURE NO. 115 Physical System Numerical Solution Governing Equations Nature Numbers Set of Mathematical Equations ERROR 1: Formulation Error ERROR 3: Data Errors ERROR 2: Numerical Errors Engineering modelers should distinguish Formulation Errors, Numerical Discretization Errors and Data Errors A MATHEMATICAL MODEL
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CE 30125 - Lecture 1 p. 1.3 Sources of Error in a Mathematical Solution • Error 1 : Missing or incorrect physics Model doesn’t include an important process (e.g. forces due to surface tension) Constitutive relationships are not a good approximation (e.g. friction law for pipes and channels not as applicable to the open ocean). • Error 2 : Numerical Solutions entail errors related to Algorithm Discretization • Boundary condition specification and domain selection Computer type • Error 3 : Observational errors occur in Measurements e.g. instruments of limited accuracy Data analysis techniques e.g. techniques may not be appropriate or based on poor or invalid assumptions and approximations Interpretation e.g. are the right things being compared?
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CE 30125 - Lecture 1 p. 1.4 Solutions to Governing Equations • It may be very difficult to solve a set of governing equations analytically (i.e. in closed form) for problems in engineering and geophysics. • Governing equations may include • Nonlinearities • Complex geometries • Varying b.c.’s • Varying material properties • Large numbers of coupled equations • These problems can not be solved analytically unless tremendous simplifications are made in the above aspects • Simplification of governing equations • Lose physics inherent to the problem • Possibly a poor answer • In general we must use numerical methods to solve the governing equations for real world problems
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CE 30125 - Lecture 1 p. 1.5 Numerical Methods
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lecture 1 - CE 30125 - Lecture 1 LECTURE 1 INTRODUCTION...

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