Unit Five: Numerical and Statistical Models

Question 1

Basis or trial functions

Answer 1

an equation selected assuming a numerical solution to a boundary value problem can be had by summing the product of a coefficent and the selected function related to a primary variable over a set number of iterations.

Question 2

Boundary methods

Answer 2

a numerical solution method for a boundary value problem that chooses functions which satisfy the governing partial differential equations exactly. The undetermined coefficients are then determined in the solution process such that the boundary conditions are approximately satisfied.

Question 3

Volume methods

Answer 3

a numerical solution method for a boundary value problem that chooses functions which satisfy the boundary conditions exactly. The undetermined coefficients are then determined in the solution process such that the governing partial differential equations are approximately satisfied.

Question 4

Collocation method

Answer 4

a method of deriving equations for undetermined coefficients where the sum of the basis functions is forced to reproduce the specified boundary conditions for boundary methods or to satisfy the governing differential equation for volume methods at a number of locations.

Question 5

Partition method

Answer 5

a method of deriving equations for undetermined coefficients where the boundary surface for boundary methods or solution domain for volume methods is partitioned into elements or subregions and the numerical solution is forced to match in the average over each element.

Question 6

Orthogonal method

Answer 6

a method of deriving equations for undetermined coefficients where 𝑁 linearly independent orthogonal functions are chosen and the difference between the numerical solution and the specified boundary conditions forced to be orthogonal to each of the functions.

Question 7

Galerkin’s method

Answer 7

a method of deriving equations for undetermined coefficients where the basis functions, Pn (x), and N linearly independent orthogonal functions, Hm (x), are taken to be the same.

Question 8

Method of least squares

Answer 8

a method of deriving equations for undetermined coefficients where the difference between the exact and approximate solutions is minimized. This technique has the advantage of converging as the number of basis functions is increased.

Question 9

Rayleigh-Ritz method

Answer 9

an analytic method for obtaining an approximate solution to a boundary value
problem, similar to Galerkin’s method, except the function to be minimized can be any function of interest.

Question 10

Finite elements

Answer 10

small discrete sections of a complex structural system to which simple elastic models can be applied, assembled together to model the complete structural system.

Question 11

Nodes

Answer 11

locations at ends, corners or along edges in a finite element model where the discrete components of the finite element model are joined together.

Question 12

Interpolation functions

Answer 12

functions used to interpolate displacements through an element given the displacements at nodes in finite element models.

Question 13

Mass matrix

Answer 13

the relation between inertia force and acceleration of an element in Finite Element Models.

Question 14

Stiffness matrix

Answer 14

the relation between force and displacement in an element in Finite Element Models.

Question 15

Sommerfeld radiation condition

Answer 15

a theorem which states that as the distance from an acoustic source approaches infinite, the acoustic pressure vanishes.

Question 16

Nonuniqueness/nonexistence

Answer 16

the phenomenon that occurs when the basis function used to generate a representation of the acoustic field of a source distribution over a surface sum to zero at certain discrete frequencies.

Question 17

Green’s function

Answer 17

a Green’s function is a mathematical tool used to solve partial differential equations. In particular it can be used in a boundary method approach to find a solution for the Helmholtz equation for various acoustics problems such as pressure distribution propagated from a monopole source.

Question 18

Pre-processor

Answer 18

a computing system element that converts engineering drawings of a structure to a grid of node locations for input to a finite element model.

Question 19

Mesh transition

Answer 19

a mesh that transitions between two different mesh sizes in a finite element model, often necessary when modelling complex structures.

Question 20

Post-processor

Answer 20

a computing system element used to generate a visual representation of the output of a finite element or boundary value analysis of a system.

Question 21

Line element

Answer 21

a component in a finite element model representing the connection between two nodes, possibly represented as a rod, a spring, or a beam.

Question 22

Surface element

Answer 22

a representation of a flat plate or curved shell component in a finite element model, formed from connecting a minimum of 3 nodes in a triangular configuration, or four nodes for a quadrangle configuration.

Question 23

Solid element

Answer 23

a four, five or six sided element with four, six or eight nodes, representing a volume in a thick structure that can be used to more accurately model the response through the thickness of the structure.

Question 24

Infinite element

Answer 24

an element introduced at unbounded edges of a finite element model to simulate an infinite model and prevent artificial reflections from the edge of the finite model.

Question 25

Strong coupling

Answer 25

a condition where undriven system components have little damping such that connected system components that are driven can easily transfer power to the undriven component, spreading the energy evenly throughout the system.

Question 26

Weak coupling

Answer 26

a condition where undriven system components have large damping such that connected system components that are driven cannot easily transfer power to the undriven component, and any energy that gets into the undriven components is quickly dissipated.

Question 27

Coupling loss factors

Answer 27

loss factors based on the connection between subsystems in a complete system, used to model the amount of power transferred between the subsystems.

Question 28

Power balance equations

Answer 28

equations used to decribe the accounting of power into and out of each subsystem to account for all power in the system.