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