1. Boundary conditions

There are many type of boundary conditions, most widely used are Natural boundary conditions, Essential boundary conditions, Dirichlet boundary conditions and Neumann boundary conditions. In a boundary value problem, what’s the difference between Essential boundary condition and Natural boundary conditions?

The two types of boundary conditions are used:

• Essential or geometric boundary conditions which are imposed on the primary variable like displacements, and
• Natural or force boundary conditions which are imposed on the secondary variable like forces and tractions.

Essential boundary conditions are conditions that are imposed explicitly on the solution and natural boundary conditions are those that automatically will be satisfied after solution of the problem. In the case of Finite Element approximations, the essential conditions will be exactly satisfied but the natural conditions only up to the order of the method. In many cases, the essential conditions correspond to Dirichlet boundary conditions when the problem is written as a boundary value problem for a partial differential equation. The natural condition corresponds to a Neumann condition, a stress-free condition, or something similar, depending on the problem. However, there are cases that are not so clear cut, so to identify $Dirichlet=essential$ and $Neumann=natural$ is not really correct. For instance, Dirichlet conditions can be assigned as natural conditions using Nitsche’s method, and Neumann conditions can be transformed to essential conditions using so-called mixed methods.

E.g. I am looking at the Ritz method for the following problem
$$
-\frac{d^{2}u}{dx^{2}}-u+x^{2}=0, 0<x<1
$$
with boundary conditions $u(0)=0$ and $\displaystyle\frac{du}{dx}\mid_{x=1}=1$.

The last derivative term, how do I know whether that is a natural or essential BC? I have googled the following guidelines but I am still confused. Specification of the primary variable ($u$ in this case) is an essential BC. Specification of a secondary variable (like a force $F$, not present in this example) is a natural boundary condition. If a boundary condition involves one or more variables in a direct way it is essential otherwise it is natural. Direct implies excluding derivative of the primary function.

As I understand the difference: what is meant is that direct gives an expression that yields a definite value for (in this case) $u$.

For example,
$$
u(0) = 0,
$$
says that at $x=0$ the value of $u$ is $0$. This is contrasted by natural expression which does not lead to a definite value of $u$.

For example,
$$
\left[ \frac{du}{dx}\right]_{x=1}=1,
$$
does not yield a definite value for $u$ at $x = 1$ since a curve of slope $1$ can be drawn through any value of $u$.

2. Local Maximum Entropy shape function

The Local Maximum Entropy shape function have a weak Kronecker-delta property at the boundary. Here the weak Kronecker-delta property at the boudary means at the boundary, the shape function value of a specified point equal to $1$ and the shape function values of other points equal to $0$. This property ensures that the values of the peformance function equal to what they should be (as user defined or as the engineering deifined) at the boundary. This is essential boundary condition/Dirichlet. So we say LME shape function enalbes the direct imposition of boundary conditions and furnishes automatic compatibility between fluids and solids or structures.