Finite Element Analysis - Quiz 1
If A and B are invertible square matrices, then ____
(AB)⁻¹=B⁻¹A⁻¹
If A is any invertible square matrix, the inverse of its inverse is the matrix A itself: Example ____
(A⁻¹)⁻¹=A
If A is any invertible square matrix, and k is any scalar, then ____
(kA)⁻¹= 1/k (A⁻¹)
If A and B are two square matrices of the same size, then A-B is defined as the sum of A+(-1)B
A+(-1)B
AA⁻¹ = I (the ___ matrix) *Similar to (x)(1/x) = 1*
Identity
MATLAB utilizes ___ and ___ algorithms
LU Decomposition Cholesky Decomposition
MATLAB provides two direct ways to solve systems of linear algebraic equations [A]{x}={b}: - ____ x = A\b - ____ x = inv(A)*b
Left-division Matrix inversion
An operation performed on a row vector (a vector transpose) and a column vector of the same size. The result is a scalar
Scalar (Dot) Product of Two Vectors
If a matrix has equal number of row and columns, it is called a ____
Square Matrix
___ operations are those that would not change the solution to a system of simultaneous linear equations, such as: - Swap position of two rows - Multiply a row by a scalar - Replace a row by sum of two rows
Valid row
C is ___ matrix of same order as A. C^T is called the ___ matrix
cofactor adjoint
For a product to be defined, the number of ___ of A must be equal to the number of ___ of B
columns rows
Matrix addition is ____ : A+B=B+A Matrix addition is ____: A+(B+C)= (A+B)+C You can also add a ____: A+0=0+A=A
commutative associative zero matrix
The ___ only exists for square matrices
determinant
If an n x n matrix H is positive definite: - All ___ elements are positive - The biggest diagonal element is bigger than the biggest ___ element - The square of any off-diagonal element, aij ,must be ___ (aiiajj)
diagonal off-diagonal less than
There are three primary methods that can be used to derive the finite element equations of a physical system. These are (1) the --- method or direct --- method for structural analysis problems, (2) the --- methods consisting of among the subsets energy methods and the principle of virtual work, and (3) the --- methods.
direct equilibrium variational weighted residual
The first treatment of two-dimensional elements was by Turner in 1956. They derived stiffness matrices for truss elements, beam elements, and two-dimensional triangular and rectangular elements in plane stress and outlined the procedure commonly known as the --- for obtaining the total structure stiffness matrix.
direct stiffness method
process of modeling a body by dividing it into an equivalent system of smaller bodies of units (finite elements) interconnected at points common to two or more elements (nodal points or nodes) and/or boundary lines and/or surfaces is called ----
discretization
An individual entry of a matrix is a ____
element
notation. The phrase --- was introduced by Clough in 1960 when both triangular and rectangular elements were used for plane stress analysis.
finite element
The --- is a numerical method for solving problems of engineering and mathematical physics. Typical problem areas of interest in engineering and mathematical physics that are solvable by use of the finite element method include structural analysis, heat transfer, fluid flow, mass transport, and electromagnetic potential.
finite element method
The finite element method involves modeling the structure using small interconnected elements called ----.
finite elements
F= Kd
global stiffness equation
If A is any square matrix and B is another square matrix satisfying the conditions: AB = BA = I Then (a) the matrix A is called invertible, and (b) the matrix B is the ___ of A and is denoted as ___. The inverse of a matrix is ___
invertible inverse A⁻¹ unique
The simplest line element (called a --- element) has two nodes, one at each end, although higher-order elements having three nodes [Figure 1-2(a)] or more (called ---, ---, etc., elements) also exist.
linear quadratic cubic
A₁, A₂→, [A₃] are three ___
matrices
A rectangular array of numbers
matrix
In MATLAB, the ____ is less efficient than left-division and also only works for square, non-singular systems.
matrix inverse
Matrix multiplication is ___ (order of addition does matter): AB≠BA in general Matrix multiplication is ___: A(BC) = (AB)C Distributive law: A(B+C) = ___, (B+C)A = ___ Multiplication by ___ matrix: AI = A; IA = A Multiplication by ___ matrix: A0 = 0;0A = 0 (AB)^T= B^TA^T
noncommutative associative AB+AC BA+CA identity zero B^TA^T
A matrix whose all entries are zero
null matrix
In the finite element method, instead of solving the problem for the entire body in ____ operation, we formulate the equations for each finite element and then combine them to obtain the solution for the whole body
one
In Quadratic form, If U>0: Matrix, K is known as ___ If U≥0: Matrix, K is known as ___
positive definite positive semidefinite
Check for negative definiteness by multiplying with -1, and then checking for ___
positive definiteness
The concept of ___ for a multi-variable function is similar to the concept of determining the curvature of a single-variable function by calculating the second derivative. In a similar way, the ___ of a function can tell us about whether the function curves upward, downward, or neither
positive definiteness Hessian Matrix
d's are called the ----
primary unknowns
If two rows or two columns are ___ (i.e. multiples of each other), then the determinant of the matrix is zero
proportional
The determinant of a matrix will be zero if: An entire ___ is zero. Two rows or columns are ___. A row or column is a constant ___ of another row or column (or a linear combination)
row equal multiple
The determinant is a ___ value and can have any value
scalar
x₁, x₂, x₃, are three ____
scalars
The determinant of a 1×1 matrix is the ____ in the matrix
single value
If the determinant is zero, the matrix is ____
singular
Inverse exists only if matrix is ___ and ___
square non-singular
the elements kij and Kij are often referred to as --- coefficients
stiffness influence
If A is a square matrix (mxm), it is called a ___ matrix if A=A^T
symmetric
The Hessian matrix is always ___
symmetric
×₁, x₂→, {x₃} are three ____
vectors
The magnitude of a vector a, |a|, is ____
√a^Ta
