Skip to main content
Ctrl+K
basics - math - Home

Linear Algebra

  • 1. Introduction to Linear Algebra
  • 2. Matrices
  • 3. Matrix factorizations
    • 3.3. Singular Value Decomposition
  • 4. Linear Systems
  • 5. Spectral decomposition
    • 5.1. Sensitivity of spectral decomposition

Multivariable Calculus

  • 6. Introduction to multi-variable calculus

Differential Geometry

  • 7. Introduction to Differential Geometry

Vector and Tensor Algebra and Calculus

  • 8. Tensor Algebra
  • 9. Tensor Calculus in Euclidean Spaces
    • 9.5. Tensor Calculus in Euclidean Spaces - Cartesian coordinates in \(E^3\)
    • 9.6. Tensor Calculus in Euclidean Spaces - cylindrical coordinates in \(E^3\)
    • 9.7. Tensor Calculus in Euclidean Spaces - Spehrical coordinates in \(E^3\)
  • 10. Time derivative of integrals over moving domains
  • 11. Calculus identities

Functional Analysis

  • 12. Introduction to Functional Analysis
  • 13. Dirac’s delta

Complex Calculus

  • 14. Complex Analysis
  • 15. Laplace Transform
  • 16. Fourier Transforms
    • 16.1. Fourier Series
    • 16.2. Fourier Transform
    • 16.3. Relations between Fourier transforms

Calculus of Variations

  • 17. Introduction to Calculus of Variations

Ordinary Differential Equations

  • 18. Introduction to Ordinary Differential Equations
  • 19. Linear Time-Invariant Systems
  • 20. LTI system response
  • 21. LTI: stability and feedback

Partial Differential Equations

  • 22. Introduction to Partial Differential Equations
  • 23. Elliptic equations
  • 24. Parabolic equations
  • 25. Hyperbolic problems
  • 26. Navier-Cauchy equations
  • 27. Navier-Stokes equations
  • 28. Arbitrary Lagrangian-Eulerian description

Numerical Methods for PDEs

  • 29. Introduction to numerical methods for PDEs
    • 29.1. Finite Element Method
    • 29.2. Finite Volume Method
    • 29.3. Boundary Element Method

Boundary Methods for PDEs

  • 30. Green’s function method

Optimization

  • 31. Optimization

Control

  • 32. Introduction to control methods
    • 32.1. Optimal control

Reinforcement Learning

  • 33. Introduction to Reinforcement Learning
  • 34. Markov Processes
  • 35. Methods of solution of MPD: DP and LP
  • 36. Methods of solution of MPD: RL
  • 37. Large or Continuous MDPs
  • .md

Navier-Cauchy equations

Contents

  • 26.1. Weak formulation

26. Navier-Cauchy equations#

Navier-Cauchy equations are the differential balance equation of the momentum of an elastic isotropic medium in the regime of small strain and displacement,

\[\rho_0 \partial_{tt} \vec{s} = \rho_0 \vec{g} + \nabla \cdot \symbf{\sigma} \ . \]

Stress tensor for an isotropic medium reads

\[\begin{split}\begin{aligned} \symbf{\sigma} & = 2 \mu \symbf{\varepsilon} + \lambda \, \text{tr} \left( \symbf{\varepsilon} \right) \mathbb{I} = \\ & = \left( 2 \mu \symbf{\varepsilon} - \frac{2}{3} \mu \, \text{tr}(\symbf{\varepsilon}) \mathbb{I} \right) + \left( \lambda + \frac{2}{3} \mu \right) \, \text{tr} \left( \symbf{\varepsilon} \right) \mathbb{I} \ , \end{aligned}\end{split}\]

with the small strain tensor

\[\symbf{\varepsilon} = \frac{1}{2} \left( \nabla \vec{s} + \nabla^T \vec{s} \right) \ .\]

Essential, natural and Robin boundary conditions read

\[\begin{split}\begin{aligned} & \vec{s} = \overline{\vec{s}} && \vec{r} \in S_D && \text{esserntial - Dirichlet b.c.} \\ & \hat{n} \cdot \symbf{\sigma} = \overline{\vec{t}}_n && \vec{r} \in S_N && \text{natural - Neumann b.c.} \\ & a \vec{s} + \hat{n} \cdot \symbf{\sigma} = \vec{b} && \vec{r} \in S_R && \text{Robin b.c.} \\ \end{aligned}\end{split}\]

26.1. Weak formulation#

For \(\forall \vec{w} \in \dots\)

\[\begin{split}\begin{aligned} 0 & = - \int_V \rho \vec{w} \cdot \partial_{tt} \vec{s} + \int_V \rho_0 \vec{w} \cdot \vec{g} + \int_{V} \vec{w} \cdot \nabla \cdot \symbf{\sigma} = \\ & = - \int_V \rho \vec{w} \cdot \partial_{tt} \vec{s} + \int_V \rho_0 \vec{w} \cdot \vec{g} + \int_{\partial V} \hat{n} \cdot \symbf{\sigma} \cdot \vec{w} - \int_{V} \nabla \vec{w} : \symbf{\sigma} \end{aligned}\end{split}\]

The volume integral containing the stress tensor can be written either as

\[\begin{split}\begin{aligned} \int_V \nabla \vec{w} : \symbf{\sigma} & = \int_V w_{i/j} \left[ \mu \left( s_{i/j} + s_{j/i} \right) + \lambda s_{k/k} \delta_{ij} \right] = \\ & = \int_V \mu w_{i/j} \left( s_{i/j} + s_{j/i} \right) + \int_V \lambda w_{j/j} s_{k/k} \end{aligned}\end{split}\]

or

\[\begin{split}\begin{aligned} \int_V \frac{1}{2} \left( \nabla \vec{w} + \nabla^T \vec{w} \right) : \symbf{\sigma} & = \int_V \frac{1}{2} \left( w_{i/j} + w_{j/i} \right) \left[ \mu \left( s_{i/j} + s_{j/i} \right) + \lambda s_{k/k} \delta_{ij} \right] = \\ & = \int_V \frac{\mu}{2} \left( w_{i/j} + w_{j/i} \right) \left( s_{i/j} + s_{j/i} \right) + \int_{V} \lambda w_{j/j} s_{k/k} \end{aligned}\end{split}\]

The weak formulation of the Navier-Cauchy equations reads

\[\int_V \rho_0 \vec{w} \cdot \partial_{tt} \vec{s} + \int_{V} 2 \mu \frac{\nabla \vec{w} + \nabla^T \vec{w}}{2} : \frac{\nabla \vec{s} + \nabla^T \vec{s}}{2} + \int_{V} \lambda \nabla \cdot \vec{w} \, \nabla \cdot \vec{s} + \int_{S_R} \vec{w} \cdot a \vec{s} = \int_{V} \rho_0 \vec{w} \cdot \vec{g} + \int_{S_N} \vec{w} \cdot \overline{\vec{t}}_n + \int_{S_R} \vec{w} \cdot \vec{b} \ ,\]

for \(\forall \vec{w} \in \dots\), and with \(\vec{s} = \overline{\vec{s}}\) for \(\vec{r} \in S_D\).

previous

25. Hyperbolic problems

next

27. Navier-Stokes equations

Contents
  • 26.1. Weak formulation

By basics

© Copyright 2022.