The polynomial requires an element with four nodes. This polynomial contains four linearly independent terms and is linear in x and y, with a bilinear term in x and y.
In order to find the three unknowns c 1 , c 2 and c 3 , we apply the boundary conditions Let us assume that the single variable can be expressed as If so, then the unknown single variable u (temperature) at any non-nodal point x, y in the 2-D domain can be expressed in terms of the known nodal variables (temperatures) u1, u 2 and u3. The single variable (for example, temperature) at these nodes 1, 2 and 3 are u1, u 2 and u3, respectively. Let us consider one such element with coordinates one such element with coordinates (x 1,y 1), (x 2,y 2), (x 3,y 3) . Each triangular element has three nodes, (i.e., one node at each corner). We consider such an area meshed with triangular elements. At each point there can be only one temperature. An example is the temperature distribution in a plate. (a scalar quantity, not a vector quantity). The physical domain considered is geometrically a 2-Dimensional domain, i.e., an area with uniform thickness and the single variable can be one of pressure, temperature, etc.
It is defined to be a state of strain in which the normal to the xy plane and the Stress ( t )directed perpendicular to the plane are assumed to be zero. It is defined to be a state of stress in which the normal stress ( s ) and shear The 2d element is extremely important for the Plane Stress analysis and Plane The basic element useful for two dimensional analysis is the = v1.rotate(angle=a, axis=vector(x,y,z)).Two dimensional elements are defined by three or more nodes in a two dimensional (0,0,1), for a rotation in the xy plane around the z axis. V2 = rotate(v1, angle=a, axis=vector(x,y,z)) There is a function for rotating a vector: Two vectors are normalized, the dot product gives the cosine of the angleīetween the vectors, which is often useful. Which is an ordinary number equal to mag(A)*mag(B)*cos(diff_angle(A,B)). The magnitude of this vector is equal mag(A)*mag(B)*sin(diff_angle(A,B)).ĭot(A,B) or A.dot(B) gives the dot product of two vectors, Hand bend from A toward B, the thumb points in the direction In a direction defined by the right-hand rule: if the fingers of the right Ĭross(A,B) or A.cross(B) gives the cross product of two vectors, a vector perpendicular to the plane defined by A and B, Magnitude, the difference of the angles is calculated to be zero. For convenience, if either of the vectors has zero To calculate the angle between two vectors (the "difference" V2.hat = v1 # changes the direction of v2 to that of v1 You can change the direction of a vector without changing its magnitude:
Norm(A) # A/|A|, normalized magnitude of 1 You can reset the magnitude to 1 with norm(): V2.mag2 = 2.7 # sets squared magnitude of v2 to V2.mag = 5 # sets magnitude to 5 no change in direction It is possible to reset the magnitude or the Vector.random() produces a vector each of whose components is a random number in the range -1 to +1 Proj(A,B) = A.proj(B) = dot(A,norm(B))*norm(B), the vector projection of A along BĬomp(A,B) = A.comp(B) = dot(A,norm(B)), the scalar projection of A along BĪ.equals(B) is True if A and B have the same components (which means that they have the same magnitude and the same direction). Norm(A) = A.norm() = A/|A|, a unit vector in the direction of the vectorĪ/|A|, a unit vector in the direction of the vector an alternative to A.norm(), based on the fact that unit vectors are customarily written in the form ĉ, with a "hat" over the vectorįor convenience, norm(vec(0,0,0)) or vec(0,0,0).hat is calculated to be vec(0,0,0).ĭot(A,B) = A.dot(B) = A dot B, the scalar dot product between two vectorsĬross(A,B) = A.cross(B), the vector cross product between two vectorsĭiff_angle(A,B) = A.diff_angle(B), the angle between two vectors, in radians Mag2(A) = A.mag2 = |A|*|A|, the vector's magnitude squared Mag(A) = A.mag = |A|, the magnitude of a vector The following functions are available for working with vectors: This is a convenient way to make a separate copy of a vector. It is okay to make a vector from a vector: vector(v2) is still vector(10,20,30). You can refer to individual components of a vector: Vectors can be added or subtracted from each other, or multiplied by an This creates a 3D vector object with the given components x, y, and z. The vector object is not a displayable object but isĪ powerful aid to 3D computations.
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