Boundary value problem: local stifness matrix

In summary, a boundary value problem is a mathematical problem used in physics, engineering, and other scientific fields to model real-world problems. A local stiffness matrix is a matrix used in finite element analysis to calculate displacements, stresses, and strains within a structure. It is important in boundary value problems as it allows for accurate modeling and calculation of structural response. The matrix is calculated by integrating over the element domain, breaking it down into sub-elements, and using shape functions and material properties. Factors such as element type, number of sub-elements, and material properties can affect the accuracy of the local stiffness matrix.
  • #1
sara_87
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Homework Statement



Given a BVP:

[tex]\Delta(u)+u=1[/tex] in [tex]\Omega[/tex]
u=0 on [tex]\partial\Omega[/tex]
using linear piecewise functions,
calculate the corresponding local stiffness matrix on the reference triangle :
{(x,y); 0<=x<=1, 0<=y<=1-x}.

The domain is a square with one point in the middle (at (0.5,0.5))

Homework Equations





The Attempt at a Solution



Does anyone know where i can start?
 
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  • #2
which numerical method are you using?
 
  • #3
I am using the finite element method.
I forgot to mention, i fixed my problem, i know how to do this question :)
 

1. What is a boundary value problem?

A boundary value problem is a type of mathematical problem that involves finding a solution to a differential equation subject to specified boundary conditions. It is commonly used in physics, engineering, and other scientific fields to model real-world problems.

2. What is a local stiffness matrix?

A local stiffness matrix is a matrix that represents the stiffness of a particular element in a finite element analysis method. It is used to calculate the displacements, stresses, and strains within a structure. The size and values of the matrix are dependent on the element type and the material properties of the structure.

3. Why is a local stiffness matrix important in boundary value problems?

A local stiffness matrix is important because it allows for the accurate modeling of complex structures and the calculation of their response to external loads. It is a crucial component in the finite element method, which is widely used in various fields of science and engineering.

4. How is a local stiffness matrix calculated?

A local stiffness matrix is calculated by integrating over the element domain and using the shape functions and material properties of the element. This process involves breaking down the element into smaller sub-elements and calculating the stiffness values for each sub-element. These values are then assembled to create the final local stiffness matrix for the element.

5. What factors can affect the accuracy of a local stiffness matrix?

The accuracy of a local stiffness matrix can be affected by various factors, such as the element type used, the number of sub-elements, and the material properties assigned to the element. Additionally, errors in the integration process or inaccurate assumptions in the model can also impact the accuracy of the local stiffness matrix.

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