• FlexPDE Help
  • Getting Started
    • Installation
    • Starting FlexPDE
    • FlexPDE Working Files
    • The Main Menu Bar
      • The File Menu
      • The Controls Menu
      • The Stop Menu
    • The Tool Bar
    • Editing Descriptor Files
    • Domain Review
    • While the Problem Runs
    • When the Problem Finishes
    • Viewing Saved Graphic Files
    • Preferences Window
      • General Settings Tab
      • Font Settings Tab
      • Web Proxy Settings Tab
      • License Registration Tab
    • Registering FlexPDE
      • Internet Key Registration
      • Dongle Registration
      • Network Dongle Registration
      • Software Key Registration
  • User Guide
    • Overview
      • What Is FlexPDE?
      • What Can FlexPDE Do?
      • How Does It Do It?
      • Who Can Use FlexPDE?
      • What Does A Script Look Like?
      • What About Boundary Conditions?
    • Basic Usage
      • How Do I Set Up My Problem?
      • Problem Setup Guidelines
      • Notation
      • Variables and Equations
      • Mapping the Domain
      • An Example Problem
      • Generating A Mesh
      • Defining Material Parameters
      • Setting the Boundary Conditions
      • Requesting Graphical Output
      • Putting It All Together
      • Interpreting a Script
    • Some Common Variations
      • Controlling Accuracy
      • Computing Integrals
      • Reporting Numerical Results
      • Summarizing Numerical Results
      • Parameter Studies Using STAGES
      • Cylindrical Geometry
        • Integrals In Cylindrical Geometry
        • A Cylindrical Example
      • Time Dependence
        • Bad Things To Do In Time Dependent Problems
      • Eigenvalues and Modal Analysis
        • The Eigenvalue Summary
    • Addressing More Difficult Problems
      • Nonlinear Coefficients and Equations
        • Complications Associated with Nonlinear Problems
      • Natural Boundary Conditions
        • Some Typical Cases
        • An Example of a Flux Boundary Condition
      • Discontinuous Variables
        • Contact Resistance
        • Decoupling
        • Using JUMP in problems with many variables
    • Using FlexPDE in One-Dimensional Problems
    • Using FlexPDE in Three-Dimensional Problems
      • The Concept of Extrusion
      • Extrusion Notation in FlexPDE
      • Layering
      • Setting Material Properties by Region and Layer
      • More Complex Extrusions
      • Void Compartments
      • Limited Regions
      • Specifying Plots on Cut Planes
      • The Complete 3D Canister
      • Setting Boundary Conditions in 3D
      • Shaped Layer Interfaces
      • Surface-Generating Functions
      • Integrals in Three Dimensions
      • More Advanced Plot Controls
      • 3D Mesh Generation
    • Complex Variables
      • The Time-Sinusoidal Heat
      • Interpreting Time-Sinusoidal Results
    • Vector Variables
      • Curvilinear Coordinates
      • Magnetic Vector Potential
    • Variables Inactive in Some Regions
      • A Chemical Beaker
    • Variables Explicit in Some Regions
    • Moving Meshes
      • Mesh Balancing
      • The Pulsating Blob
      • Mesh Reconnection
    • Controlling Mesh Density
    • Post-processing with FlexPDE
    • Exporting Data to Other Applications
    • Importing Data from Other Applications
    • Using ARRAYS and MATRICES
    • Solving Nonlinear Problems
    • Using Multiple Processors
    • Running FlexPDE from the Command Line
    • Running FlexPDE Without A Graphical Interface
    • Importing DXF Files
    • Importing OBJ Files
    • Getting Help
  • Problem Descriptor Reference
    • Introduction
      • Preparing a Descriptor File
      • File Names and Extensions
      • Problem Descriptor Structure
      • Problem Descriptor Format
      • Case Sensitivity
      • "Include" Files
      • A Simple Example
    • The Elements of a Descriptor
      • Comments
      • Reserved Words and Symbols
      • Separators
      • Literal Strings
      • Numeric Constants
      • Built-in Functions
        • Analytic Functions
        • Non-Analytic Functions
        • Unit Functions
        • String Functions
        • The FIT Function
        • The LUMP Function
        • The RAMP Function
        • The SAVE Function
        • The SUM Function
        • The SWAGE Function
        • The VAL and EVAL functions
        • Boundary Search Functions
      • Operators
        • Arithmetic Operators
        • Complex Operators
        • Differential Operators
        • Integral Operators
          • Time Integrals
          • Line Integrals
          • 2D Surface Integrals
          • 3D Surface Integrals
          • 2D Volume Integrals
          • 3D Volume Integrals
        • Relational Operators
        • String Operators
        • Vector Operators
        • Tensor Operators
      • Predefined Elements
      • Expressions
      • Repeated Text
    • The Sections of a Descriptor
      • Title
      • Select
        • Mesh Generation Controls
        • Solution Controls
        • Global Graphics Controls
      • Coordinates
      • Variables
        • The THRESHOLD Clause
        • The ORDER Clause
        • Complex Variables
        • Moving Meshes
        • Variable Arrays
        • Vector Variables
      • Global Variables
      • Definitions
        • ARRAY Definitions
        • MATRIX Definitions
        • COMPLEX Definitions
        • Function Definitions
        • STAGED Definitions
        • POINT Definitions
        • TABLE Import Definitions
          • The TABLE Input function
          • The TABLEDEF input statement
          • TABLE Modifiers
          • TABLE File format
        • TABULATE definitions
        • TRANSFER Import Definitions
        • The PASSIVE Modifier
        • Mesh Control Parameters
      • Materials
      • Boundary Conditions
      • Initial Values
      • Equations
        • Association between Equations, Variables and Boundary Conditions
        • Sequencing of Equations
        • Initial Equations
        • Eigenvalues and Modal Analysis
        • Moving Meshes
      • Constraints
      • Extrusion
      • Rotation
      • Boundaries
        • Points
        • Boundary Paths
        • Regions
          • Reassigning Regional Parameters
          • Regions in One Dimension
          • Regions in Three Dimensions
          • Regional Parameter Values in 3D
          • Limited Regions in 3D
          • Empty Layers in 3D
        • Excludes
        • Features
        • Node Points
        • Ordering Regions
        • Numbering Regions
        • Fillets and Bevels
        • Specifying Boundary Conditions
          • Syntax of Boundary Condition Statements
          • Point Boundary Conditions
          • Boundary conditions in 1D
          • Boundary Conditions in 3D
          • Jump Boundaries
          • Periodic Boundaries
          • Complex and Vector Boundary Conditions
      • Front
      • Maximize
      • Minimize
      • Resolve
      • Time
      • Monitors and Plots
        • Graphics Display Specifications
        • Graphic Display Modifiers
        • Controlling the Plot Domain
        • Reports
        • The ERROR Variable
        • Window Tiling
        • Monitors in Steady State Problems
        • Monitors and Plots in Time Dependent Problems
        • Hardcopy
        • Graphics Export
        • Data Export
          • Data Export Specifications
          • TRANSFER File format
          • TRANSFER6 File format
        • Examples
      • Histories
      • End
    • Batch Processing
  • Technical Notes
    • Applications in Electromagnetics
      • Introduction
        • Finite Element Methods
        • Principles
        • Boundary Conditions
        • Integration by Parts and Natural Boundary Conditions
        • Adaptive Mesh Refinement
        • Time Integration
        • Summary
      • Electrostatics
        • Electrostatic Fields in 2D
        • Electrostatics in 3D
        • Capacitance per Unit Length in 2D Geometry
      • Magnetostatics
        • A Magnet Coil in 2D Cylindrical Coordinates
        • Nonlinear Permeability in 2D
        • Divergence Form
        • Boundary Conditions
        • Magnetic Materials in 3D
      • Waveguides
        • Homogeneous Waveguides
        • TE and TM Modes
        • Non-Homogeneous Waveguides
        • Boundary Conditions
        • Material Interfaces
      • References
    • Natural Boundary Conditions
    • Solving Nonlinear Problems
    • Avoid Discontinuities!
    • Smoothing Operators in PDE's
    • Interpreting Error Estimates
    • Coordinate Scaling
    • Making Movies
    • Converting from Version 4 to Version 5
    • Converting from Version 5 to Version 6
    • Converting from Version 6 to Version 7
    • How FlexPDE Works
  • Sample Problems
    • Applications
      • Chemistry
        • chemburn
        • melting
        • reaction
      • Control
        • control_steady
        • control_transient
      • Electricity
        • 3d_capacitor
        • 3d_capacitor_check
        • 3d_dielectric
        • capacitance
        • dielectric
        • fieldmap
        • plate_capacitor
        • space_charge
      • Fluids
        • 1d_eulerian_shock
        • 1d_lagrangian_shock
        • 2d_eulerian_shock
        • 2d_piston_movingmesh
        • 3d_flowbox
        • airfoil
        • black_oil
        • buoyant
        • buoyant+time
        • cavity_1k
        • channel
        • contaminant_transport
        • coupled_contaminant
        • coupled_contaminant_initeq
        • flowslab
        • geoflow
        • hyperbolic
        • landfill_gas_flow
        • lowvisc
        • swirl
        • viscous
      • Groundwater
        • porous
        • richards
        • water
      • Heatflow
        • 1d_float_zone
        • 3d_bricks
        • 3d_bricks+time
        • axisymmetric_heat
        • float_zone
        • heat_boundary
        • radiation_flow
        • radiative_boundary
        • slider
      • Lasers
        • laser_heatflow
        • self_focus
      • Magnetism
        • 3d_helmholtz_coil
        • 3d_magnetron
        • 3d_vector_magnetron
        • magnet_coil
        • permanent_magnet
        • saturation
      • Misc
        • diffusion
        • minimal_surface
        • surface_fit
      • Quantun_Mechanics
        • 1d_finite_potential_well
        • 2d_finite_potential_well
        • morse_potential
      • Stress
        • 3d_bimetal
        • anisotropic_stress
        • axisymmetric_stress
        • bentbar
        • bentbar_moving
        • elasticity
        • fixed_plate
        • free_plate
        • harmonic
        • plasticity
        • prestube
        • tension
        • vibrate
    • Usage
      • 1D
        • 1d_cylinder
        • 1d_cylinder_transient
        • 1d_float_zone
        • 1d_slab
        • 1d_sphere
      • 3D_domains
        • 2d_sphere_in_cylinder
        • 3d_box_in_sphere
        • 3d_cocktail
        • 3d_cylspec
        • 3d_ellipsoid
        • 3d_ellipsoid_shell
        • 3d_extrusion_spec
        • 3d_fillet
        • 3d_guitar
        • 3d_helix_layered
        • 3d_helix_wrapped
        • 3d_lenses
        • 3d_limited_region
        • 3d_pinchout
        • 3d_pipes
        • 3d_planespec
        • 3d_pyramid
        • 3d_shell
        • 3d_shells
        • 3d_sphere
        • 3d_spherebox
        • 3d_spherespec
        • 3d_spool
        • 3d_thermocouple
        • 3d_toggle
        • 3d_torus
        • 3d_torus_tube
        • 3d_twist
        • 3d_void
        • off_axis_disks_2
        • off_axis_disks_3
        • regional_surfaces
        • tabular_surfaces
        • two_spheres
        • twoz_direct
        • twoz_import
        • twoz_export
        • twoz_planar
      • 3D_Rotations
        • Rotated_heat_ring
      • Accuracy
        • forever
        • gaus1d
        • gaus2d
        • gaus3d
        • rotating_shape
        • sine1d
        • sine2d
        • sine3d
      • Arrays+Matrices
        • arrays
        • array_boundary
        • matrices
        • matrix_boundary
        • wave_boundary
      • CAD_Import
        • arrow_OBJimport
        • boxinbox_OBJimport
        • canincan_OBJimport
        • cube_OBJimport
        • cube2_OBJimport
        • cube2_validate
        • cubes_bigsmall_OBJimport
        • cubes_bigsmall_validate
        • gourd_OBJimport
        • helix_OBJimport
        • horn_OBJimport
        • torus_OBJimport
        • tubefaces_OBJimport
        • wineglass_OBJimport
      • Constraints
        • 3d_constraint
        • 3d_surf_constraint
        • boundary_constraint
        • constraint
      • Contact_Boundaries
        • 3d_contact
        • 3d_contact_region
        • contact_resistance_heating
        • thermal_contact_resistance
        • transient_contact_resistance_heating
      • Coordinate_Scaling
        • scaled_z
        • unscaled_z
      • Eigenvalues
        • 3d_oildrum
        • 3d_plate
        • drumhead
        • drumhole
        • drumhole_animate
        • filledguide
        • shiftguide
        • vibar
        • waveguide
        • waveguide20
      • Functions
        • complex_functions
        • floor_ceiling
        • function_definition
        • error_reporting
        • globalmax
        • globalmax_3d
        • standard_functions
        • sum
        • swage_pulse
        • swage_test
        • unit_functions
        • vector_functions
      • Implicit_Curves
        • implicit_curve_boundary
        • implicit_curve_surface
        • sine_boundary
        • sine_boundary_3d
      • Import_Export
        • 3d_mesh_export
        • 3d_mesh_import
        • 3d_post_processing
        • 3d_surf_export
        • blocktable
        • export
        • export_format
        • export_history
        • mesh_export
        • mesh_import
        • post_processing
        • smoothtable
        • splinetable
        • table
        • tabledef
        • table_export
        • table_import
        • transfer_export
        • transfer_import
      • Integrals
        • 2d_integrals
        • 3d_integrals
        • time_integral
      • Mesh_Control
        • 3d_curvature
        • boundary_density
        • boundary_spacing
        • front
        • mesh_density
        • mesh_spacing
        • resolve
      • Misc
        • askuser
        • bc_sets
        • bc_switching
        • data_fitting
        • edge_distance
        • fillet
        • fit+weight
        • ifthen
        • lump
        • material_sets
        • moving_source
        • point_load
        • polar_coordinates
        • regional_equations
        • repeat
        • rotated_ellipse
        • save
        • spacetime1
        • spacetime2
        • spline_boundary
        • staged_geometry
        • stages
        • tabulate
        • tensors
        • two_histories
      • Moving_Mesh
        • 1d_stretch_x
        • 2d_blob_position
        • 2d_blob_velocity
        • 2d_bubble_reconnect
        • 2d_lagrangian_shock
        • 2d_movepoint
        • 2d_stretch_x
        • 2d_stretch_xy
        • 3d_blob_position
        • 3d_blob_velocity
        • 3d_bubble_reconnect
        • 3d_intrusion_reconnect
        • gaussian_valve_reconnect
        • sliptube_explicit_reconnect
      • ODE
        • linearode
        • nonlinode
        • second_order_time
      • Optimization
        • criticality
        • criticality_size
        • power_control
        • size_control
      • Periodicity
        • 1d_periodic
        • 2d_wave_propagation
        • 3d_antiperiodic
        • 3d_periodic_exchange
        • 3d_xperiodic
        • 3d_zperiodic
        • antiperiodic
        • azimuthal_periodic
        • periodic+time
        • periodic
        • periodic_contaminant
        • periodic_map
        • three-way_periodic_3d
        • two-way_periodic
        • two-way_periodic_3d
      • Plotting
        • 3d_ploton
        • export_test
        • plot_on_grid
        • plot_test
      • Sequenced_Equations
        • equation_iteration
        • initialeq
        • smoothing_discontinuities
        • theneq+time
        • theneq
      • Stop+Restart
        • restart_export
        • restart_import
      • Variable_Types
        • array_variables
        • complex_emw21
        • complex_sinusoidal_heat
        • complex_variables
        • complex+time
        • inactive_variables
        • vector_lowvisc
        • vector_variables
        • vector+time
  • Release Notes