# COMSOL Multiphysics Glossary

This glossary contains terms related to finite element modeling, mathematics, geometry, and CAD as they relate to the COMSOL Multiphysics software. For more application-specific terms, see the glossaries in the AC/DC Module, Acoustics Module, CAD Import Module, CFD Module, Chemical Reaction Engineering Module, Heat Transfer Module, MEMS Module, RF Module, and Structural Mechanics Module.

The glossary content is copyright © COMSOL, Inc. For permission to reproduce content, contact .

A method of improving solution accuracy by adapting the mesh to the problem’s physical behavior.

### affine transformations

Geometric transformations that are combinations of linear transformations and translations.

### algebraic multigrid (AMG)

An algebraic multigrid solver or preconditioner 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).

### anisotropy

Variation of material properties with direction.

### application programming interface (API)

An API provides a set of documented functions and methods for interacting with a software product.

### arbitrary Lagrangian-Eulerian formulation (ALE formulation)

A formulation for a moving mesh where dependent variables represent the mesh displacement or mesh velocity. The COMSOL Multiphysics solvers have built-in support for the mesh movement.

### arc

A segment of the circumference of a circle or ellipse.

### Argyris element

A 2D, 6-node triangular finite element with a 5th-order basis function providing continuous derivatives between elements.

### aspect ratio

The ratio between the longest and shortest element or geometry dimension.

### assemble

Taking the local element stiffnesses, masses, loads, and constraints to form the stiffness matrix, mass matrix, load vector, constraint matrix, and constraint residual vector.

### associative geometry

An algorithm that maps data associated with a geometry to the new geometry entities when the geometry is modified.

### backward differentiation formula (BDF)

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.

### basis function

A function in the finite element space such that the ith degree freedom is 1, while all other degrees of freedom are 0. For the Lagrange finite element space, is a linear or higher order polynomial on each mesh element with value 1 in node i and 0 in all other nodes.

### Bernstein polynomial

See Bézier basis.

### Bézier basis

A set of polynomial functions that occur in the definition of a Bézier curve. These polynomial functions are often called Bernstein polynomials.

### Bézier curve

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.

### Bézier patch, Bézier surface

A Bézier patch or Bézier surface is a surface extension of the Bézier curve. The Bézier patch is a function of two variables with an array of control points.

See constraint.

### Boolean operations

Boolean operations are used to construct a geometry object from other solid geometry objects and rebuild it in a new form. 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:

• Union (add): the resultant geometry object occupies all the space of the initial geometry objects
• Difference (subtract): the resultant geometry object occupies all the space of the first geometry object except for the space inside the second geometry object.
• Intersection: the resultant geometry object occupies only the space common to the initial geometry objects

### boundary

A geometric entity with a space dimension one less than the space dimension for the geometry (for example, a face in a 3D 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.

### border

The interface between two parts in an assembly.

### boundary modeling

A geometry modeling method to create a geometry by defining its boundaries. Compare to solid modeling and surface modeling.

### brick element

See hexahedral element.

### chamfer

A CAD feature that trims off a corner with a plane or straight line.

### Cholesky factorization

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.

### coefficient form PDE

A PDE in the coefficient form is a PDE formulation suited for linear PDEs

### composite geometry object, composite solid object

Geometric objects made up by combining primitive geometry objects and other composite objects. See also constructive solid geometry, primitive geometry object, and Boolean operations.

### COMSOL Desktop

The integrated simulation environment for the COMSOL Multiphysics products with a number of windows such as the Model Builder window, the Graphics window, and the Settings window.

### COMSOL Multiphysics binary file

A binary data file with the extension `.mphbin` that contains geometry objects or mesh objects. Earlier versions of COMSOL Multiphysics used the file extension `.flb`.

### COMSOL Multiphysics text file

A text data file with the extension `.mphtxt` that contains geometry objects or mesh objects.

### condition number

A measure of the possible error in a solution due to ill-conditioning of the equations. See also ill-conditioning.

### constant

A named model property that has a constant numeric value.

### constraint

Restriction imposed upon the dependent variables, typically as a Dirichlet boundary condition. Neumann boundary conditions are not regarded as constraints. When Dirichlet boundary conditions are introduced, the finite element algorithm makes a corresponding change to the Neumann boundary conditions so that the resulting model becomes solvable. For a bidirectional constraint, COMSOL Multiphysics accomplishes this change by adding the transpose of the constraint matrix h times a vector of Lagrange multipliers to the right-hand side of the Neumann boundary condition. For a unidirectional constraint, the extra term is often some matrix times the vector of Lagrange multipliers. In a mechanical model, the extra term is called a constraint force.

### constructive solid geometry (CSG)

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.

### control point

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.

### control weight

Scalar values assigned to control points to further control the shape of a curve or surface.

### contour plot

A plot that shows the variation of a solution component or other quantity. Points with equal values of the plotted quantity are connected with contour lines.

### convergence

The tendency for a finite element solution to approach the exact solution within well-defined and specified tolerances, for example, by reducing the mesh element size or the time step.

### coupling operator

An operator used to couple data within a model (geometry) or between different models (geometries). See also extrusion coupling operator, projection coupling operator, and integration coupling operator. Coupling variables provided similar functionality in earlier version of COMSOL Multiphysics, but coupling operators can be reused with different arguments (for example, for integrating different quantities over the same domain).

### curl element

See vector element.

### curve

The path of a point moving through space. See also Bézier curve, NURBS, and manifold.

### curve object

A geometry object consisting of only edges and vertices, for example a geometry object representing a curve.

### curve segment

An individual polynomial or rational polynomial curve. Compounded curves consist of several curve segments.

### degree of freedom (DOF)

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.

### deformed geometry

A geometry where the shape changes with a moving-mesh algorithm. It also the name of a physics interface for modeling deforming geometries. This is similar to the Parameterized Geometry physics interface in earlier versions of COMSOL Multiphysics.

### dependent variable

A varying quantity whose changes are arbitrary, but they are regarded as produced by changes in other variables. For example, temperature is a function of the space coordinates and time. In a narrower sense, the dependent variables, or solution components, are the unknowns in a mathematical PDE model. Compare to independent variable.

### differential-algebraic equation (DAE)

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.

### direct solver

A solver for a system of linear equation that uses some variant of Gaussian elimination. Compare to iterative solver.

### Dirichlet boundary condition

A Dirichlet boundary condition specifies the value of the function (dependent variable) on a boundary. Dirichlet boundary conditions are sometimes called essential boundary conditions or constraints. For a coefficient form PDE the Dirichlet boundary condition is

.

### discretization

The process of dividing a continuous system into a finite number of elements with finite size. The difference between the finite-element representation and the real system, the discretization error, drops as the size of the elements decrease. For a time-dependent analysis, a discretization of time into steps provides an idealized behavior of the variations in the solution during these steps.

### divergence element

A finite element often used for electromagnetic vector fields. The degrees of freedom on the boundary of a mesh element correspond to normal components of the field. Also Nédélec’s divergence element.

### domain

A topological part of the modeling space in a geometry model. The geometric representation of a domain is a line segment (interval) in 1D, an area in 2D, and a volume in 3D. In a mathematical context, the symbol Ω represents the domain where the equations are defined.

### drop tolerance

A nonnegative scalar used in the incomplete LU preconditioner for the iterative solvers. See incomplete LU factorization.

### dynamic model

See time-dependent model.

### edge, edge segment

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.

### edge element

See vector element.

### eigenvalue PDE

A PDE that describes an eigenvalue problem with unknown eigenmodes (eigenfunctions) u and eigenvalues λ. The coefficient form eigenvalue PDE is:

### elliptic PDE

A linear stationary 2nd-order elliptic PDE has the form

where c is positive or negative definite, for example, Poisson’s equation.

### embed

To insert a 2D geometry into a 3D geometry model.

### error

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).

### error estimate

An estimation of the error in the numeric solution to a problem, either locally or globally, primarily for use by an adaptive mesh refinement. See also adaptive mesh refinement, error.

### equivalent boundaries

Boundaries that are rigid transformations of each other and have compatible meshes. See also periodic boundary condition.

### essential boundary condition

See Dirichlet boundary condition.

### expression variable

A user defined variable that is defined on any geometry domain in terms of dependent variables, independent variables, constants, application scalar variables, and other expression variables. Global expression variables are valid in all geometries; scalar expression variables are valid in the current geometry.

### extended mesh

A data structure that includes the full finite element mesh. See also mesh, node point.

### extended multiphysics

A model that includes nonlocal couplings and dependencies between variables, where the value at a point is the result of a computation elsewhere in the domain or in another geometry defined in the same model. Coupling variables provide the ability to project or extrude values from one geometry or domain to another. Compare to multiphysics.

### exterior boundary

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 exterior of the computational domain. See also boundary.

### extrude

To create a 3D geometry object from a 2D geometry object in a work plane by translating (extruding) it along a path, often a straight line.

### extruded mesh

A 3D mesh created by extrusion of a 2D mesh. An extruded mesh can contain hexahedral elements and prism elements.

### extrusion coupling operator

An operator in the destination that takes values from the source by interpolation at points that depend on the position of the evaluation points in the destination.

### face, face segment

A domain describing a bounded part of a surface in a 3D geometry. A face or face segment is a boundary in a 3D geometry. See also domain.

### face object

A geometry object with no topological information on domains. Typically a trimmed surface is represented as a face object.

### FEM

See finite element method.

### FEM structure

The main data structure, containing all data for a model.

### fillet

A curved transition from one boundary to another, creating a rounded corner.

### finite element

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 Multiphysics generally uses finite element in the mathematical sense.

### finite element analysis (FEA)

A computer-based analysis method for field problems using the finite element method.

### finite element method (FEM)

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.

### finite element space

The linear space of functions where the finite element approximation to the solution of a PDE problem is sought. The functions in the finite element space are linear combinations of basis functions (shape functions).

### flux vector

The vector . See also generalized Neumann boundary condition and normal flux.

### free mesh

An unstructured mesh that can represent any geometry. Compare to mapped mesh.

### free mesher

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.

The mesh generator creating unstructured quadrilateral meshes.

### free triangle mesher

The mesh generator creating unstructured triangular meshes.

### Gauss point

Sometimes improperly used as a synonym for integration point. See also integration point.

### general form PDE

A PDE in the general form is a PDE formulation suited for nonlinear PDEs

### generalized Neumann boundary condition

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 generalized Neumann condition is often called just Neumann condition in the documentation.

### geometric multigrid (GMG)

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).

### geometry level

The types of topological entities within a geometry model that describes bounded parts of the manifolds in the model, and also the relations between different manifolds in the geometry. The different geometry levels are the vertex, edge, face, and domain levels. An entity of dimension one less than the space dimension is referred to as a boundary. See also manifold.

### geometry model

A collection of topological and geometric entities that form a complete geometric description of the model.

### geometry object

The objects that represent a geometry model. See also curve object, face object, primitive geometry object, solid object.

### grid

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.

### Hermite element

A finite element similar to the Lagrange element. The difference is that there are degrees of freedom for the (1st-order) space derivatives at the mesh vertices. See also Lagrange element.

### hexahedral element

A 3D mesh element with eight corners and six faces, also referred to as brick element; sometimes also called hex element as a short form.

### higher-order element

A finite element with basis functions that consists of polynomials of degree 2 or higher.

### hybrid geometry modeling

Creating a geometry model using a combination of boundary modeling/surface modeling and solid modeling.

### hyperbolic PDE

A typical example of a linear 2nd-order hyperbolic PDEs is the wave equation

where ea and c are positive.

### IGES file

An IGES file contains 3D CAD data, including the 3D geometry, in an open format according to the Initial Graphics Exchange Specification. You can import an IGES file to COMSOL Multiphysics using the CAD Import Module.

### ill-conditioning

An ill-conditioned system is sensitive to small changes in the inputs and is susceptible to roundoff errors. See also condition number.

### imprint

An imprint of the 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 a curve on the boundary in 3D.

### incomplete LU factorization

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.

### independent variable

A variable that can cause variation in a second, dependent variable. The independent variables are most often space coordinates and time. Compare to dependent variable.

### index, for DAE

See differential-algebraic equation.

### index vector

See domain group.

### initial condition

The starting values for the dependent variables in a time-dependent analysis and for nonlinear iterations or other iterative solvers.

### integration coupling operator

An operator that evaluates integrals of expressions over the source and returns a single scalar value when used in the destination. Similar functionality is available to evaluate the average, minimum, and maximum values.

### integration point

See numerical integration formula.

### interactive meshing

Building a mesh in an incremental fashion where each meshing operation acts on a set of geometry domains.

### interior boundary

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.

### interval

The domain between two vertices in a 1D geometry. Also called a domain.

### isoparametric element

A finite element that uses the same shape function for the element shape coordinates as for the dependent variables.

### isosceles triangle

A triangle with at least two equal sides (and two equal angles).

### iteration

See iterative solver.

### iterative solver

A solver for a system of linear equations that uses an iterative method, computing 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.

### Jacobian matrix

A matrix containing the first derivative of a vector-valued function of a vector variable. In particular, it is the derivative of the residual vector with respect to the solution vector. When used in this narrower sense, the term stiffness matrix is sometimes used.

### Lagrange element

A finite element with polynomial shape functions of a certain order. The value of the function is used as the degree of freedom, and the node points are evenly distributed within the mesh element.

### Lagrange multiplier

An extra dependent variable introduced in the Neumann boundary condition when a constraint is added. See also constraint.

### linear iteration

A step in a linear iterative solver. See iterative solver. Compare to nonlinear iteration.

### linear PDE

An equation where both sides are sums of a known function, the unknown functions, and their partial derivatives, multiplied by known coefficients that only depend on the independent variables. Other PDEs are called nonlinear.

### LU factorization

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.

### manifold

A mathematical function describing a surface, curve, or point in a geometry model of any dimension.

### mapped mesh

A structured mesh with quadrilateral elements generated by mapping using transfinite interpolation.

### mapped mesher

The mesh generator creating mapped meshes.

### mass matrix

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).

### mesh

A subdivision of the domains of a geometric model into, for example, triangles (2D) or tetrahedra (3D). These are examples of mesh elements. See also grid, structured mesh, and unstructured mesh.

### mesh element

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.

### mesh vertex

An endpoint or corner of a mesh element. See also node point and vertex.

### method of lines

A method for solving a time-dependent PDE through a space discretization, resulting in a set of ODEs.

### mixed boundary condition

See generalized Neumann boundary condition.

### mode reduction

A model-reduction technique for reducing systems with many degrees of freedom, for example large finite element models, to a form with fewer degrees of freedom for dynamic system simulations and analysis. See also state-space model.

### model input

Model inputs are fields such as temperature and velocities that act as inputs for material models and model equations. The model inputs can be fields computed by other physics interfaces or user defined values.

### Model MPH-file

A binary data file with the extension `.mph` that contains a COMSOL Multiphysics model. Often also just called model file. Earlier versions of COMSOL Multiphysics used the file extension `.fl`.

### Model M-file

An M-file containing commands that create a COMSOL Multiphysics model. Model M-files are text files that can be modified and used with MATLAB. If you have a MATLAB license and a license for the COMSOL LiveLink for MATLAB, the COMSOL Desktop can load a Model M-file. Compare with Model MPH-file.

### MRI data

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.

### multidisciplinary models

Multidisciplinary models combine PDE-based finite element modeling with other mathematical modeling techniques such as dynamic simulation in areas like automatic control and signal processing.

### multigrid

A solver or preconditioner for a linear system of equations that computes a sequence of increasingly accurate approximations of the solution by using a hierarchy of coarsened versions of the linear system (having fewer degrees of freedom). See also algebraic multigrid, geometric multigrid.

### multiphysics

Multiphysics models include more than one equation and variable from different types of physics. These variables can be defined in different domains. The equations can be coupled together through equation coefficients that depend on variables from other equations. Compare to extended multiphysics.

### natural boundary condition

See Neumann boundary condition.

### 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.

### Newton’s method

An iterative solver method, also called the Newton-Raphson method, for solving nonlinear equations. See also nonlinear iterations.

### Newton-Raphson method

See Newton’s method.

### node point

Any point in the mesh element where the degrees of freedom are defined. The node points often include the mesh vertices and possible interior or midpoint locations. See also degree of freedom (DOF) and mesh vertex.

### nonlinear iteration

A Newton step in the solution of a nonlinear PDE problem. Each nonlinear iteration involves the solution of a linear system of equations. Compare to linear iteration.

See linear PDE.

### norm

A scalar measure of the magnitude of a vector or a matrix. Several types of norms are used to measure the accuracy of numerical solutions.

### numerical integration formula

A numeric-integration method that approximates an integral by taking the weighted sum of the integrand evaluated at a finite number of points, the integration points (sometimes improperly called Gauss points). Also called quadrature formula.

### normal flux

The normal component of the flux vector at a boundary.

### NURBS

The nonuniform rational B-spline (NURBS) is a popular curve and surface representation scheme. A NURBS representation can be divided into a number of rational Bézier curves or surfaces.

### order of a finite element

The degree of the polynomials that define the shape functions (basis functions).

### ordinary differential equation (ODE)

An equation involving functions and their derivatives. The derivatives are with respect to one independent variable only. Compare to partial differential equation (PDE).

### parabolic PDE

A typical example of a linear 2nd-order parabolic PDE is the heat equation

where da and c are positive.

### parameter

A constant that takes on different values for each model in a parametric analysis. See also constant.

### partial differential equation (PDE)

An equation involving functions and their partial derivatives; that is, an equation that includes derivatives with respect to more than one independent variable. Compare to ordinary differential equation (ODE).

### periodic boundary condition

A boundary condition where the values of the solution appear in a periodic pattern, typically so that the value of the solution on one boundary is equal to the value on another boundary. See also equivalent boundaries.

### phasor

A complex number or a vector of complex numbers representing a sinusoidally varying current or voltage.

### physics interfaces

Sets of features for different types of physics in the COMSOL Desktop environment. The physics interfaces contain predefined equations and boundary conditions and a set of features for setting up models for a certain type of physics. The application modes in earlier versions of COMSOL Multiphysics provides similar functionality.

### pivot

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 term when it is used to eliminate values below it is called the pivot value.

### point

A location in space.

### point object

A geometry object with only vertices.

### positive definiteness

A symmetric matrix is positive definite when all its eigenvalues are positive.

### preconditioner

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.

### primitive, primitive geometry object

A geometry object with a basic shape such as a cube or a sphere. You can add primitives to a model, using arbitrary sizes and positions, and combine them to form complex shapes. See also constructive solid geometry, composite geometry object, and Boolean operations.

### prism element

A 3D mesh element with six corners and five faces, also referred to as wedge element.

### projection coupling operator

An operator that takes values from the source by evaluating line integrals over lines whose positions are dependent on the position of the evaluation points in the destination.

See numerical integration formula.

A 2D mesh element with four corners and four edges; sometimes also called quad element as a short form.

### rational Bézier curve

See Bézier curve.

### residual vector

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.

### revolve

To create a 3D geometry object from a 2D geometry object in a work plane by rotating it around an axis.

### revolved mesh

A 3D mesh created by revolving a 2D mesh. A revolved mesh can contain hexahedral elements and prism elements.

### Robin boundary condition

See generalized Neumann boundary condition.

### shape function

A basis function described in local element coordinates. See also basis function.

### shift

A value σ around which an eigensolver searches for eigenvalues.

### simplex element

Triangle element in 2D and tetrahedral element in 3D.

### solid

A description of a part of the modeling space. See also domain.

### solid modeling

A 3D geometry modeling method that describes both the boundary and interior of the geometry using solid objects. See also constructive solid geometry (CSG) and solid.

### solid object

A geometry object representing one or several solids.

### solution component

See dependent variable.

### solution matrix

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. See also solution structure.

### solution structure

A data structure that includes the solution vector or solution matrix and any associated data such as parameter values, output times, or eigenvalues.

### solution vector

A vector with all the degrees of freedom (values of the dependent variables) as its components. See also solution matrix and solution structure.

### solver sequence

A recorded sequence of named solver settings and commands that can be replayed by a single solver call.

### sparse matrix

Matrix for which the number of zero elements is large enough to justify special data types and algorithms that avoid operations on zero elements.

### split

To divide a geometry object into its minimal parts.

### stability

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.

### state-space model

A linear time-invariant representation of a dynamic system as a set of 1st-order ODEs of the form

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.

### static model

See stationary model.

### stationary model

A model where the dependent variables do not change over time. It typically represents a steady-state solution. Also called static model or steady model.

See stationary model.

### stiffness matrix

See Jacobian matrix.

### streakline

The locus of particles that have earlier passed through a prescribed point in space. See also streamline.

### streamline

A curve that is everywhere tangent to the vector field (in particular a velocity field) at a given instant of time. Sometimes called a flow line or flux line. See also streakline.

### streamline-diffusion stabilization

A numerical technique for stabilization of the numeric solution to a PDE by artificially adding diffusion in the direction of the streamlines.

### strong form

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.

### structured mesh

A mesh for which all elements and nodes have the same topology. Compare to unstructured mesh.

### surface

A mathematical function (manifold) from 2D to 3D space.

### surface normal

A vector perpendicular to the surface.

### surface modeling

A 3D geometry modeling method to describe a geometry by defining its bounding surfaces. Compare boundary modeling and solid modeling.

### swept mesh

A 3D mesh generated by sweeping a face mesh along a domain.

### symmetric matrix

A matrix that equals its own transpose.

### symmetric successive overrelaxation (SSOR)

A symmetric successive overrelaxation (SSOR) preconditioner uses classic SSOR iterations.

### symmetry

The invariance of an object attribute or of the object itself under a well-defined operation or 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. Axisymmetry is a common type of symmetry.

### symmetry boundaries

See equivalent boundaries.

See weak form.

### tetrahedral element

A 3D mesh element with four corners, six edges, and four triangular faces.

### time-dependent model

See transient model.

### transient model

A model where at least one of the dependent variables changes over time, for example, the heat equation or the wave equation. Also called dynamic model, time-dependent model, or unsteady model.

### triangular element

A 2D mesh element with three corners and three edges.

### trimmed surface

If the parameter space of a surface is divided into “valid” and “invalid” regions, the image of the valid regions is called the trimmed surface. This corresponds to the part of the surface limited by a closed loop of edges lying on the surface.

See constraint.

### unstructured mesh

A mesh without a specific pattern where the elements can have different shapes and the nodes can have different connectivities. Compare to structured mesh.

See time-dependent model.

### vector element

A finite element often used for electromagnetic vector fields. Each mesh element has degrees of freedom corresponding only to tangential components of the field. Also called curl element, Nédélec’s edge element, or just edge element.

### vertex

A point in a geometry model, often an endpoint of a geometry segment or an intersection of geometry entities of 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.

### weak constraint

A reformulation of a Dirichlet boundary condition as a weak form equation. When using a weak constraint, the corresponding Lagrange multiplier becomes a solution component (dependent variable).

### weak form

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. Most physics interfaces in COMSOL Multiphysics are implemented using a weak form. Compare to strong form.

### wedge element

See prism element.

### work plane

An embedded 2D work space that can be positioned relative to the coordinate planes or an already existing geometry. Using work planes makes it possible to define a geometry in terms of previously created geometry objects such as points, edges, and faces.