An algebraic multigrid solver or preconditioner that performs one or more cycles of a multigrid method using a coarsening of the discretization based on the coefficient matrix. Compare to
geometric multigrid (GMG).
An API provides a set of documented functions and methods for interacting with a software product.
A 2D, 6-node triangular finite element with a 5th-order basis function providing continuous derivatives between elements.
Taking the local element stiffnesses, masses, loads, and constraints to form the stiffness matrix,
mass matrix, load vector, constraint matrix, and constraint residual vector.
A multistep formula based on numerical differentiation for solutions to ordinary differential equations. A BDF method of order
n computes the solution using an
nth-grade polynomial in terms of backward differences.
A function i in the
finite element space such that the
ith degree of freedom is 1, while all other degrees of freedom are 0. For the Lagrange finite element space,
i is a linear or higher-order polynomial on each mesh element with value 1 in node
i and 0 in all other nodes.
See Galerkin boundary element method.
A set of polynomial functions that occur in the definition of a Bézier curve. These polynomial functions are often called
Bernstein polynomials.
A rational Bézier curve is a parameterized
curve formed as the quotient of two polynomials expressed in the Bézier basis. It is a vector-valued function of one variable. The coefficients of a rational Bézier curve are geometrically interpreted as
control points and
control weights. A
nonrational Bézier curve is a rational Bézier curve with all weights equal, thereby making the denominator polynomial equal to a constant. A nonrational Bézier curve is also called an
integer Bézier curve.
A Bézier patch or
Bézier surface is a surface extension of a
Bézier curve. A
Bézier patch is a function of two variables with an array of control points.
A constraint enforced by reaction terms affecting both equations in a constraint of the type u1 = u2.
Symmetric constraints are an important special case. See also
reaction terms and
constraint.
Boolean operations are used to construct a geometry object from other geometry objects. At least two primary geometry objects are required to create a resultant new geometry object. That new object depends on the type of Boolean operation:
A geometric entity with a dimension one less than the space dimension for the geometry (a
face in a 3D geometry, an
edge in a 2D geometry, and a
vertex in a 1D geometry). In a mathematical context, the symbol
∂Ω represents the boundary of the domain
Ω . Sometimes
boundary is used in a narrower sense meaning an
exterior boundary. See also
interior boundary,
exterior boundary.
See Galerkin boundary element method.
A memory-saving version of LU factorization where
U is the transpose of
L. It requires that the coefficient matrix
A (
A =
LU) be a symmetric positive definite matrix. See also
LU factorization and
positive definiteness.
Geometric objects made up by combining primitive geometry objects and other composite objects. See also
constructive solid geometry,
primitive geometry object, and
Boolean operations.
For each application launched from a COMSOL Server ™ installation, a separate COMSOL Application Server process is started to run the application. A COMSOL Application Server process contains functionality that is similar to a
COMSOL Multiphysics Server together with functionality to generate the application’s user interface accessed from a web browser or a
COMSOL Client.
A Windows® client that runs an implementation of a COMSOL application, created with the Application Builder, and that connects to a
COMSOL Server™.
The COMSOL Desktop® is an integrated simulation environment for the COMSOL
products with a number of windows such as the Model Builder window, the Graphics window, and each model tree node’s Settings window.
A binary data file with the extension .mphbin that contains geometry objects or mesh objects.
A text data file with the extension .mphtxt that contains geometry objects or mesh objects.
Restriction imposed upon the dependent variables on the form R(u1,u2,...) = 0. A
Dirichlet boundary condition is a special case.
Neumann boundary conditions are not regarded as constraints. When a constraint is added, the finite element algorithm adds corresponding
reaction terms to the system of equations. These generalized
reaction forces modify the
flux conditions so that the resulting model becomes solvable.
A solid-modeling method that combines simple solid shapes, or primitives, to build more complex models using Boolean operations. See also
solid modeling and
primitive.
A boundary condition or source is contributing when it adds to other boundary conditions or sources defined on the same geometric entity. Examples of contributing boundary conditions are loads in structural mechanics and heat flux components in heat transfer. See also
exclusive nodes.
Bézier and
NURBS curves and surfaces are defined by a set of points known as
control points. The locations of these points control the curve’s shape.
Scalar values assigned to control points to further control the shape of a curve or surface.
A geometry object consisting of only edges and
vertices (where no vertex is isolated), for example, a geometry object representing a
curve.
One of the unknowns in a discretized finite element model. A degree of freedom is defined by a name and a node point. The degree of freedom names often coincide with the names of the dependent variables. The local degrees of freedom are all degrees of freedom whose node points are in one mesh element.
A geometry where the shape changes with a moving-mesh algorithm. It is also the name of a physics interface for modeling deforming geometries. This is similar to the Parameterized Geometry interface in earlier versions of COMSOL Multiphysics.
A set of equations that includes both differential and algebraic equations. A DAE is classified in terms of its index, a positive integer, which is related to the minimum number of differentiations needed to transform a DAE to an ODE form.
See time-dependent model.
A geometric entity representing a bounded part of a
curve. An
edge or
edge segment is a
boundary in a 2D geometry. See also
domain.
where c is positive or negative definite (for example, Poisson’s equation).
Deviations from the correct solution, primarily due to: poor modeling; discretization (such as insufficiently fine mesh, poor elements, or insufficiently short time steps); and roundoff and truncation (depending on numerical representation,
ill-conditioning, or the solution algorithms).
Boundaries that are rigid transformations of each other and have compatible meshes. See also
periodic boundary condition.
See Dirichlet boundary condition.
An Eulerian formulation means that the partial differential equations that describe some physics are formulated in a spatial frame (coordinate system), with coordinate axes fixed in space. An Eulerian formulation is common for fluid flow when the focus is on specific locations in space through which fluid flows. Compare to
Lagrangian formulation.
A boundary condition or material model in a domain is exclusive when there can only be one such node defined for a given geometric entity. Adding another exclusive boundary condition to the same boundary, for example, the last added boundary condition (last in the Model Builder tree) overrides any other similar boundary condition defined on the same boundary. Examples of exclusive boundary conditions are prescribed displacements in structural mechanics and specified temperature in heat transfer. See also
contributing node.
An exterior boundary for a dependent variable
u is a
boundary such that
u is defined only on one of the adjacent domains, that is, a boundary to the computational domain. See also
boundary.
To create a 3D geometry object from a 2D geometry object in a work plane or a planar face in 3D by translating (extruding) it in the normal direction.
A geometric entity describing a bounded part of a
surface in a 3D geometry. A
face is a
boundary in a 3D geometry. See also
domain.
See finite element method.
The resulting geometry used for assigning materials and physics. COMSOL Multiphysics creates the finalized geometry by forming a union of the entire geometry sequence or by forming an assembly where the geometry objects in the geometry sequence are treated as individual parts. The finalized geometry consists of
geometric entities.
In the mathematical sense, a mesh element together with a set of
shape functions and corresponding
degrees of freedom. The linear combinations of the shape functions form a space of functions called the
finite element space. In the traditional FEA sense, the concept of a finite element also includes the discretized form of the PDEs that govern the physics. COMSOL generally uses
finite element in the mathematical sense.
A computational method that subdivides an object into very small but finite-size elements. The physics of one element is approximately described by a finite number of degrees of freedom (DOFs). Each element is assigned a set of characteristic equations (describing physical properties, boundary conditions, and imposed forces), which are then solved as a set of simultaneous equations to predict the object’s behavior.
A computation method that, in ways similar to the finite element method, computes values at discrete places on a meshed geometry. Finite volume refers to the small volume surrounding each node point in a mesh.
A boundary condition that specifies the value of the normal flux across a boundary, also known as a
natural boundary condition. A (generalized)
Neumann boundary condition is a special case.
A frame is a coordinate system that is fixed in space, to a material, to the geometry, or to a mesh. The frames make it possible to use an
Eulerian formulation or a
Lagrangian formulation for various physics in a model or using the arbitrary Lagrangian-Eulerian (ALE) method. The following frame types are available:
material frame (
reference frame),
geometry frame,
mesh frame, and
spatial frame.
An unstructured mesh that can represent any geometry. Compare to
mapped mesh.
The mesh generator creating free meshes. The mesh generator creating
triangular elements is also referred to as the
free triangle mesher, and the mesh generator creating
quadrilateral elements is also referred to as the
free quad mesher.
COMSOL Multiphysics supports user-defined functions, which can be analytic, piecewise, and interpolation functions as well as special types of common functions that implement, for example, steps, ramps, and other wave forms. There are also common built-in
mathematical functions such as trigonometric functions, logarithms, and special functions.
A Gauss point is an integration point in the special case of numerical integration using Gaussian quadrature. Sometimes,
Gauss point is improperly used as a synonym for
integration point. See also
integration point.
A generalized Neumann boundary condition (also called a mixed boundary condition or a
Robin boundary condition) specifies the value of a linear combination of the
normal flux and the
dependent variables on a boundary. For a coefficient form PDE, the generalized Neumann boundary condition is
The basic parts that constitute the finalized geometry: In 3D, they are divided into the following four types or geometric entity levels:
domains,
boundaries (
faces),
edges, and
points (
vertices). In 2D, there are no
faces, and the
edges are the
boundaries. In 1D, there are only
domains and
points, which are also the
boundaries.
The geometric entity levels are the
vertex,
edge,
face, and
domain levels (in 3D). An entity of dimension one less than the space dimension is referred to as a
boundary. See also
geometric entities.
A geometric multigrid solver or preconditioner performs one or more cycles of a multigrid method, using a coarsening of the discretization based on a coarsening of the mesh or a reduction in the order of the shape functions. Compare to
algebraic multigrid (AMG).
In the geometry frame (coordinate system), the domain is fixed and identical to the original geometry. No physics is formulated directly in the geometry frame — only the material frame and
spatial frame have physical significance. The geometry frame is used only as a reference for the Deformed Geometry interface and for postprocessing. When there is no Deformed Geometry interface present, the geometry frame is identical to the material frame.
A collection of geometric entities that form a complete geometric description of the model.
An object generated by a geometry feature. See also point object,
curve object,
surface object,
primitive geometry object,
solid object, and
mixed object.
A grid usually refers to sets of evenly-spaced parallel lines at particular angles to each other in a plane, or the intersections of such lines. Compare to
mesh.
A finite element similar to the Lagrange element. The difference is that there are degrees of freedom for the (first-order) space derivatives at the mesh vertices. See also
Lagrange element.
A finite element with basis functions that consists of polynomials of degree 2 or higher.
Creating a geometry model using a combination of boundary modeling/
surface modeling and
solid modeling.
where ea and
c are positive.
An IGES file contains 3D CAD data, including the 3D geometry, in an open format according to the
Initial Graphics Exchange Specification. IGES files can be imported into COMSOL Multiphysics using the CAD Import Module.
An imprint of the usually smaller boundary on the larger boundary that makes the parts in a pair match. An imprint inserts points on the boundary in 2D and creates edges on the boundary in 3D.
An approximate LU factorization where small matrix elements are discarded to save memory and computation time. The
drop tolerance is a relative measure of the smallness of the elements that should be discarded. See also
LU factorization.
A variable that can cause variation in a second, dependent variable. The independent variables are most often spatial coordinates and time. Compare to
dependent variable.
See differential-algebraic equation.
The order (a positive integer) in a numerical integration formula where evaluation points and weights are chosen such that result of the numerical integration is exact for all expressions that are polynomials of an order that is no higher than the specified
integration order.
See numerical integration formula.
An interior boundary for a dependent variable
u is a
boundary such that
u is defined on both adjacent domains or in no adjacent domain. See also
boundary.
An inverted curved element occurs when a
curved mesh element inverts locally when more node points are added to better approximate the shape of the geometry. There are many possible geometrical causes, but the element is often too large compared to the geometry feature size.
A finite element that uses the same shape function for the element shape coordinates as for the
dependent variables.
A solver for a system of linear equations that uses an iterative method, calculating a sequence of more and more accurate approximations to the solution. Each step in this sequence is one linear iteration. This should not be confused with the Newtons iterations (
nonlinear iterations) that occur in the solution of a nonlinear system of equations. Compare to
direct solver and
nonlinear iteration.
A finite element with polynomial shape functions of a certain
order (degree). The value of the function is used as the
degree of freedom, and the node points are the
Lagrange points.
An extra dependent variable introduced in the flux conditions when a constraint is added. The Lagrange multiplier often has a physical meaning and an interpretation as a (generalized)
reaction force. See also
constraint.
In a mesh element, the Lagrange points of order
k are the points whose local (element) coordinates are integer multiples of
1/
k. These points are used as node points for the
Lagrange element. For example, the Lagrange points of order 1 are the corners of the mesh element.
A Lagrangian formulation means that the partial differential equations that describe some physics are formulated in a material frame (coordinate system) with coordinate axes fixed to the material in its reference configuration and following the material as it deforms. The Lagrangian formulation is common for solid mechanics because it makes anisotropic material properties independent of the current spatial orientation of the material. Compare to
Eulerian formulation.
A step in a linear iterative solver. See iterative solver. Compare to
nonlinear iteration.
For a linear system of equations, a version of Gaussian elimination that produces a factorization A = LU of the coefficient matrix, where
L and
U are the lower and upper triangular matrices, respectively. This makes it easy to quickly solve a number of systems with the same coefficient matrix. See also
direct solver.
A structured mesh with
quadrilateral elements generated by mapping using transfinite interpolation.
The matrix E that multiplies the second time derivative of the
solution vector in the linearized discretized form of a PDE problem. If there are no second time derivatives (that is, if
E = 0), then the term mass matrix is often used for the matrix
D that multiplies the first derivative of the solution vector (the
D matrix is otherwise called the
damping matrix).
Built-in common mathematical constants such as π and
i and numerical constants such as the machine precision or machine epsilon.
The individual elements in the mesh that together form a partitioning of the geometry, for example, triangular elements and
tetrahedral elements. See also
finite element. A
curved mesh element is a mesh element that is extended with additional node points to better approximate the shape of the geometry.
See generalized Neumann boundary condition.
A nonempty geometry object that is not a
solid object,
surface object,
curve object, or
point object. For example, the union of a solid object and a curve object is a mixed object.
Model inputs are fields such as temperature and velocities or other physical properties that act as inputs for material properties and model equations. The model inputs can be fields computed by other physics interfaces or user-defined values.
A file that contains Java® commands calling on the COMSOL
API. Use a text editor to extend and modify the model file. Compiling and running a model file for Java creates the COMSOL Multiphysics model.
A text file containing commands that create a COMSOL Multiphysics model. A model file for MATLAB is a text file (M-file) that is similar to a model file for Java and that can be modified and used with MATLAB. If you have a MATLAB license and a license for LiveLink™
for MATLAB®, the COMSOL Desktop can load a model file for MATLAB. Compare with
Model MPH-file.
A binary data file with the extension .mph that contains a COMSOL Multiphysics model or application. Often also just called model file or application file.
Magnet resonance imaging (MRI) data is an image data format, primarily for medical use. MRI produces high-quality images of the inside of the human body. 3D MRI data is usually represented as a sequence of 2D images.
See Neumann boundary condition.
A Neumann boundary condition specifies the value of the normal flux across a boundary. Neumann boundary conditions are sometimes called
natural boundary conditions. Compare to
generalized Neumann conditions.
An iterative solver method, also called the Newton-Raphson method, for solving nonlinear equations. See also
nonlinear iterations.
User-defined nonlocal couplings are used to couple data within a model component (geometry) or between different model components (geometries). See also extrusion nonlocal coupling,
projection nonlocal coupling, and
integration nonlocal coupling. Nonlocal couplings can be reused with different arguments (for example, for integrating different quantities over the same domain).
The normal component of the flux vector at a boundary.
The nonuniform rational B-spline (NURBS) is a curve and surface representation scheme. A NURBS representation can be divided into a number of
rational Bézier curves or surfaces.
In the context of model reduction, online refers to using the reduced order for evaluations rather than producing them.
A user-defined operator function, or just
operator, is similar to a
function but behaves differently. For example, COMSOL Multiphysics includes differentiation operators that take expressions as input arguments to define a derivative of an expression with respect to a variable. There are also built-in arithmetic, relational, and logical operators.
where da and
c are positive.
Usually a value on the main diagonal of the stiffness matrix.
Pivoting is the interchanging of rows and columns in order to place a particularly large element in the diagonal position. The value of the diagonal element when it is used to eliminate values below it is called the
pivot value.
A symmetric matrix is positive definite when all its eigenvalues are positive.
The convergence rate of iterative methods depends on the spectral properties of the coefficient matrix. A preconditioner is a matrix that transforms the linear system into one that has the same solution but that has more favorable spectral properties. See also
algebraic multigrid,
geometric multigrid,
incomplete LU factorization,
iterative solver, and
SSOR.
See numerical integration formula.
Terms that are automatically added to the system of equations in order to enforce a constraint. Reaction terms from boundary constraints appear as a
flux condition and share the same physical meaning. Using an analogy from structural mechanics, reaction terms are sometimes referred to as (generalized) reaction forces.
The vector L in the discretized form of a PDE problem. In the absence of
constraints, the discrete form of a stationary equation is
0 = L(
U) where
U is the
solution vector.
See generalized Neumann boundary condition.
A basis function described in local element coordinates. See also
basis function.
A value σ around which an eigensolver searches for eigenvalues.
Triangle element in 2D and
tetrahedral element in 3D.
A geometry object whose vertices,
edges, and
faces all have an adjacent
domain.
A matrix that contains a sequence of solutions as columns. A steady-state problem results in a solution vector, but eigenvalue problems, time-dependent problems, and parametric analyses produce a
solution matrix.
A vector with components that contain all the degrees of freedom (values of the
dependent variables) as its components. See also
solution matrix.
A solver for a time-dependent model is unconditionally stable if the initial conditions are not amplified artificially and the roundoff errors do not grow, regardless of the size of the time step. A solver is
conditionally stable if there is a maximum value of the time step above which the numerical solution is unstable.
where x is the state vector,
u is the input, and
y is the output.
A,
B,
C, and
D are the constant dynamics, input, output, and direct transmission matrices, respectively.
A partial differential equation in the strong form is the standard formulation as an equality of functions. The strong form is divided into the
coefficient form and the
general form. Compare to
coefficient form,
general form, and
weak form.
A symmetric successive overrelaxation (SSOR) preconditioner uses classic SSOR iterations.
The invariance of an object attribute or of the object itself under a transformation such as inversion, rotation, or reflection. A symmetry allows for a reduction of the model geometry so that appropriate boundary conditions account for the redundant portions of the geometry. Axial symmetry is a common type of
symmetry.
A constraint that is enforced by
reaction terms chosen so as to preserve the symmetry of symmetric unconstrained systems. This choice of reaction terms is unique and leads to a
bidirectional constraint that modifies the equations corresponding to all dependent variables appearing in the constrained expression.
See equivalent boundaries.
A constraint enforced by reaction terms that only affect one of the dependent variables in a constraint of type
u1 = u2. The other dependent variables are treated as independent with respect to the unidirectional constraint. Compare to
symmetric constraint. See also
constraint.
See time-dependent model.
A point in a geometry model, often an endpoint of an edge or an intersection of geometric entities of a higher degree such as
edges or
faces. A vertex is referred to as a
point for the specification of point sources and other PDE modeling. See also
domain.
A reformulation of a constraint as a
weak form equation. When using a weak constraint, the corresponding
Lagrange multiplier becomes a
solution component (
dependent variable).
A partial differential equation in the weak form is a more general formulation than the strong form. It is produced by multiplying the
strong form PDE with an arbitrary function called the
test function and integrating over the computational domain. Physics interfaces in COMSOL Multiphysics are implemented using a weak form. Compare to
strong form.
An embedded 2D work space that can be positioned relative to the coordinate planes or an already existing 3D geometry. Using work planes makes it possible to define a geometry in terms of previously created geometry objects such as
points,
edges, and
faces. From a work plane with a 2D geometry, 3D geometry objects can be created using
extrude or
revolve operations.