Boundary conditions are defined in the mesh creation step in the hopr.ini file and can be modified when running PICLas in the corresponding parameter.ini file. In the hopr.ini file, which is read by the hopr executable, a boundary is defined by
BoundaryName = BC_Inflow ! BC index 1 (from position in the parameter file)
BoundaryType = (/4,0,0,0/) ! (/ Type, curveIndex, State, alpha /)
where the name of the boundary is directly followed by its type definition, which contains information on the BC type, curving, state and periodicity. This can be modified in the parameter.ini file, which is read by the piclas executable via
BoundaryName = BC_Inflow ! BC name in the mesh.h5 file
BoundaryType = (/5,0/) ! (/ Type, State /)
In this case the boundary type is changed from 4 (in the mesh file) to 5 in the simulation.
The boundary conditions used for Maxwell's equations are defined by the first integer value in the BoundaryType vector (consisting of the Type and State) and include, periodic, Dirichlet, Silver-Mueller, perfectly conducting, symmetry and reference state boundaries as detailed in the following table.
| (/Type,State/) | Type | State |
|---|---|---|
| (/1,1/) | periodic | 1: positive direction of the 1st periodicity vector |
| (/1,-1/) | periodic | -1: negative (opposite) direction of the 1st periodicity vector |
| (/2,2/) | Dirichlet | 2: Coaxial waveguide |
| (/2,22/) | Dirichlet | 22: Coaxial waveguide BC (boundary condition or exact flux) |
| (/2,3/) | Dirichlet | 3: Resonator |
| (/2,4/) | Dirichlet | 4: Electromagnetic dipole (implemented via RHS source terms and shape function deposition) |
| (/2,40/) | Dirichlet | 40: Electromagnetic dipole without initial condition (implemented via RHS source terms and shape function deposition) |
| (/2,41/) | Dirichlet | 41: Pulsed Electromagnetic dipole (implemented via RHS source terms and shape function deposition) |
| (/2,5/) | Dirichlet | 5: Transversal Electric (TE) plane wave in a circular waveguide |
| (/2,7/) | Dirichlet | 7: Special manufactured Solution |
| (/2,10/) | Dirichlet | 10: Issautier 3D test case with source (Stock et al., div. correction paper), domain [0;1]^3 |
| (/2,12/) | Dirichlet | 12: Plane wave |
| (/2,121/) | Dirichlet | 121: Pulsed plane wave (infinite spot size) and temporal Gaussian |
| (/2,14/) | Dirichlet | 14: Gaussian pulse is initialized inside the domain (usually used as initial condition and not BC) |
| (/2,15/) | Dirichlet | 15: Gaussian pulse with optional delay time tDelayTime |
| (/2,16/) | Dirichlet | 16: Gaussian pulse which is initialized in the domain and used as a boundary condition for t>0 |
| (/2,50/) | Dirichlet | 50: Initialization and BC Gyrotron - including derivatives |
| (/2,51/) | Dirichlet | 51: Initialization and BC Gyrotron - including derivatives (nothing is set for z>eps) |
| (/3,0/) | SM | 1st order absorbing BC (Silver-Mueller) - Munz et al. 2000 / Computer Physics Communication 130, 83-117 with fix |
| of div. correction field for low B-fields that only set the correction fields when ABS(B)>1e-10 | ||
| (/5,0/) | SM | 1st order absorbing BC (Silver-Mueller) - Munz et al. 2000 / Computer Physics Communication 130, 83-117 |
| (/6,0/) | SM | 1st order absorbing BC (Silver-Mueller) - Munz et al. 2000 / Computer Physics Communication 130, 83-117 with fix |
| of div. correction field for low B-fields that only set the correction fields when B is significantly large compared to E | ||
| (/4,0/) | PEC | Perfectly electric conducting surface (Munz, Omnes, Schneider 2000, pp. 97-98) |
| (/10,0/) | Symmetry | Symmetry BC (perfect MAGNETIC conductor, PMC) |
| (/20,0/) | Ref | Use state that is read from .h5 file and interpolated to the BC |
Dielectric -> type 100?
The boundary conditions used for Maxwell's equations are defined by the first integer value in the BoundaryType vector (consisting of the Type and State) and include, periodic, Dirichlet (via pre-defined function, zero-potential or RefState), Neumann and reference state boundaries as detailed in the following table.
| (/Type,State/) | Type | State |
|---|---|---|
| (/1,1/) | periodic | 1: positive direction of the 1st periodicity vector |
| (/1,-1/) | periodic | -1: negative (opposite) direction of the 1st periodicity vector |
| (/2,0/) | Dirichlet | 0: Phi=0 |
| (/2,2/) | Dirichlet | 2: Automatic adjustment for Phi to meet const. input power, see {ref}sec:fixed-coupled-power
|
| (/2,1001/) | Dirichlet | 1001: linear potential y-z via Phi = 2340y + 2340z |
| (/2,101/) | Dirichlet | 101: linear in z-direction: z=-1: 0, z=1, 1000 |
| (/2,103/) | Dirichlet | 103: dipole |
| (/2,104/) | Dirichlet | 104: solution to Laplace's equation: Phi_xx + Phi_yy + Phi_zz = 0 |
| (/2,200/) | Dirichlet | 200: Dielectric Sphere of Radius R in constant electric field E_0 from book: John David Jackson, Classical Electrodynamics |
| (/2,300/) | Dirichlet | 300: Dielectric Slab in z-direction of half width R in constant electric field E_0: |
| adjusted from CASE(200) | ||
| (/2,301/) | Dirichlet | 301: like CASE=300, but only in positive z-direction the dielectric region is assumed |
| (/2,400/) | Dirichlet | 400: Point Source in Dielectric Region with |
| epsR_1 = 1 for x |
||
| epsR_2 != 1 for x |
||
| (/4,0/) | Dirichlet | zero-potential (Phi=0) |
| (/5,1/) | Dirichlet | 1: use RefState Nbr 1 and sec:ref-state-bcs) |
| (/6,1/) | Dirichlet | 1: use RefState Nbr 1 and sec:ref-state-bcs) |
| (/7,1/) | Dirichlet | 1: use LinState Nbr 1, linear function for Phi, see {ref}sec:linear-potential
|
| (/8,1/) | Dirichlet | 8: Assign BC to EPC group nbr. 1 (different BCs can be assigned the same EPC), see {ref}sec:electric-potential-condition
|
| (/10,0/) | Neumann | zero-gradient (dPhi/dn=0) |
| (/11,0/) | Neumann | q*n=1 |
| (/20,1/) | FPC | 1: Assign BC to FPC group nbr. 1 (different BCs can be assigned the same FPC), see {ref}sec:floating-boundary-condition
|
| (/50,0/) | Dirichlet | {ref}sec:bias-voltage-for-dc (0: this number has no meaning) |
| (/51,1/) | Dirichlet | {ref}sec:bias-voltage-for-ac (1: use RefState Nbr 1, see {ref}sec:ref-state-bcs) |
| (/52,1/) | Dirichlet | {ref}sec:bias-voltage-for-ac-and-cpp (1: use RefState Nbr 1, see {ref}sec:ref-state-bcs) and {ref}sec:fixed-coupled-power
|
| (/60,1/) | Dirichlet | {ref}sec:fixed-coupled-power for sec:ref-state-bcs) |
(sec:ref-state-bcs)=
For each boundary of type 5 (reference state boundary RefState), e.g., by setting the boundary in the parameter.ini file
BoundaryName = BC_WALL ! BC name in the mesh.h5 file
BoundaryType = (/5,1/) ! (/ Type, State/)
the corresponding RefState number must also be supplied in the parameter.ini file (here 1) and is selected from its position in the parameter file. Each RefState is defined in the parameter.ini file by supplying a value for the voltage an alternating frequency for the cosine function (a frequency of 0 results in a fixed potential over time) and phase shift
RefState = (/-0.18011, 1.0, 0.0/) ! RefState Nbr 1: Voltage, Frequency and Phase shift
This yields the three parameters used in the cosine function
where A=-0.18011 is the amplitude, t is the time, f=1 is the frequency and psi=0 is the phase shift.
Similar to boundary type 5 is type 6, which simply uses a cosine function that always has the same sign, depending on the amplitude A
(sec:linear-potential)=
A linear function that ramps the electric potential from 0 V to a user-defined value can be applied to a boundary via
BoundaryName = BC_WALL ! BC name in the mesh.h5 file
BoundaryType = (/7,1/) ! 1: 1st LinState
Additionally, this specific boundary condition requires a starting position LinPhiBasePoint and
a direction along which the potential varies LinPhiNormal. The distance along which the potential varies as well as the final
value are defined by LinPhiHeight and LinPhi, respectively.
Coordinates below and above this distance are simply set to 0 V and the defined value, respectively.
The example below creates a linear ramp from 0 V to 1000 V starting at 1 mm in z-direction and ramps the value over 10 mm in the
same direction.
LinPhiBasePoint = (/0. , 0. , 1e-3/)
LinPhiNormal = (/0. , 0. , 1.0/)
LinPhiHeight = 10e-3
LinPhi = 1000.
The linear potential uses the same functionality as RefState, hence, when two different functions are to be defined use the following example
BoundaryName = BC_right
BoundaryType = (/7,1/) ! 7: Dirichlet with linear ramp 1st LinState
LinPhiBasePoint = (/0. , 0. , 0./) ! 1st LinState
LinPhiNormal = (/1. , 0. , 0./) ! 1st LinState
LinPhiHeight = 1.0 ! 1st LinState
LinPhi = 1e3 ! 1st LinState
BoundaryName = BC_left
BoundaryType = (/7,2/) ! 7: Dirichlet with linear ramp 2nd LinState
LinPhiBasePoint = (/0. , 0. , 0./) ! 2nd LinState
LinPhiNormal = (/1. , 0. , 0./) ! 2nd LinState
LinPhiHeight = 1.0 ! 2nd LinState
LinPhi = 0.0 ! 2nd LinState
(sec:floating-boundary-condition)=
A floating boundary condition (FPC) can be used to model a perfect electric conducting surface. The surface can carry a charge
BoundaryName = BC_FPC_1 ! BC name in the mesh.h5 file
BoundaryType = (/20,1/) ! 20: activate FPC, 1: Index of the FPC group to which this BC belongs (1st group)
BoundaryName = BC_FPC_2 ! BC name in the mesh.h5 file
BoundaryType = (/20,2/) ! 20: activate FPC, 2: Index of the FPC group to which this BC belongs (2nd group)
For this boundary condition, the charge assigned to each FPC and the resulting potential are written to FieldAnalyze.csv automatically, e.g., "007-FPC-Charge-BCState-001" and "008-FPC-Voltage-BCState-001", where BCState corresponds to the ID of the FPC. If the particle boundary condition is set to open (or species-swap), then each impacting charged particle that is removed there, will be added to the accumulated charge on that FPC.
(sec:electric-potential-condition)=
Grounded surfaces with a specific resistance between the ground and the surface can be modelled as equipotential surfaces, where charged particles are removed and their charge is accumulated over each time step and a voltage is calculated from from the resistance of the surface and the resulting electric current via
where
BoundaryName = BC_EPC_1 ! BC name in the mesh.h5 file
BoundaryType = (/8,1/) ! 8: activate EPC, 1: Index of the EPC group to which this BC belongs (1st group)
BoundaryName = BC_EPC_2 ! BC name in the mesh.h5 file
BoundaryType = (/8,2/) ! 8: activate EPC, 2: Index of the EPC group to which this BC belongs (2nd group)
The resulting current
(sec:fixed-coupled-power)=
An automatic adjustment of a DC electric potential to ensure that a fixed power absorbed by the charged particles of the system is achieved, requires the following parameters
BoundaryName = BC_WALL ! BC name in the mesh.h5 file
BoundaryType = (/2,2/) ! all BCs with this type will be adjusted to the same electric potential that is adjusted over time
For an AC boundary the parameters are the following
BoundaryName = BC_WALL ! BC name in the mesh.h5 file
BoundaryType = (/60,1/) ! applicable to one BC with this BCType. The BCState=1 corresponds to the RefState number
Additionally, a starting value for the potential
CoupledPowerPotential = (/10. , 1000. , 2000./) ! lower, starting and maximum values for the electric potential
CoupledPowerRelaxFac = 0.05 ! the new potential is updated by 5% in each time step
CoupledPowerTarget = 1e-10 ! target power of 1e-10 Watt
The values in CoupledPowerPotential correspond to the lower boundary, the starting value and the upper boundary, respectively.
When a simulation is restarted from a state file, the last known value of the BC will be used instead of the starting value, which
is only applied when starting a fresh simulation from scratch.
Note that in the case of an AC boundary, the amplitude defined in the RefState will be overwritten with the initial value given in
CoupledPowerPotential.
There are three models implemented by which the power control adjusts the electric potential (or peak potential in case of AC)
CoupledPowerMode = 3 ! 1: instantaneous power (default value), 2: moving average, 3: integrated and averaged over 1 cycle
CoupledPowerFrequency = 13.56e6 ! Frequency with which the coupled power voltage is adapte
where CoupledPowerMode selects the model and CoupledPowerFrequency defines the adaption frequency and corresponding period over
which the power is integrated but only when model 3 is used. Model 1 uses the instantaneously determined absorbed power that inherently
oscillates, model 2 uses an averaged value (that is reset during every load balance or normal restart event) and model 3 integrated
the power linearly between particle analysis steps and adjusts the power after each period of the user-defined frequency
CoupledPowerFrequency.
It is important to note that when no Dirichlet boundary conditions are selected by the user, the code automatically enforces mixed
boundaries on either Neumann or periodic boundaries. Depending on the simulation domain, the direction with the largest extent is
selected and on those boundaries an additional Dirichlet boundary condition with
| Zero potential side activated in direction (1: x, 2: y, 3: z) | 1 | OUTPUT |
To selected the direction by hand, simply supply the desired direction via
HDGZeroPotentialDir = 1
with 1: x-, 2: y-, 3: z-direction.
(sec:bias-voltage-for-dc)=
A bias voltage develops if charged particles impact on an electrode that is connected via capacitor to a matching box. Hence, if
there is an overhead of electrons impacting on the electrode, a negative bias voltage arises, which counters the flow of electrons
to that boundary. This feature only works when PARTICLES=ON is enabled and Poisson's equation is solved PICLAS_EQNSYSNAME=poisson.
The following parameters are required
UseBiasVoltage = T ! Activate bias voltage model
BiasVoltage-NPartBoundaries = 2 ! Nbr of particle boundaries where the ion excess is measured
Biasvoltage-PartBoundaries = (/1,2/) ! Particle boundary index of where the ion excess is measured
BiasVoltage-Frequency = 3e9 ! Frequency with which the bias voltage is adapted
BiasVoltage-Delta = 1.0 ! Voltage difference used for adapting the bias voltage
where UseBiasVoltage switches the modell on/off, BiasVoltage-NPartBoundaries defines the number of boundaries over which the
total electric charge is summarized, Biasvoltage-PartBoundaries specifies the indices of the particle boundaries where the total
electric charge is summarized, e.g., the powered and the grounded electrode., BiasVoltage-Frequency is the frequency with which
the bias voltage is updated by the voltage difference BiasVoltage-Delta.
Additionally, the field boundary must be defined
BoundaryName = BC_left ! BC name in the mesh.h5 file
BoundaryType = (/50,0/) ! Dirichlet with 0V initial BC
where the value 50 is used for for pure DC boundaries.
Furthermore, the bias voltage model relies on the functionality of the integral particle flux measurement on the corresponding
boundaries, see BoundaryParticleOutput (BPO) in Section {ref}sec:integral-surface-variables.
The definition of BPO variables must include at least the ones above, e.g.
CalcBoundaryParticleOutput = T
BPO-NPartBoundaries = 2
BPO-PartBoundaries = (/1,2/)
BPO-NSpecies = 3
BPO-Species = (/2,3,4/)
where the defined species information must consider all relevant charged particle species that affect the bias voltage.
(sec:bias-voltage-for-ac)=
If an AC potential boundary is coupled with a bias potential, the BoundaryType has to be changed as compared with the previous
section
BoundaryName = BC_left ! BC name in the mesh.h5 file
BoundaryType = (/51,1/) ! Dirichlet with 0V initial BC
Furthermore, a RefState must be defined, which specifies the parameters for the sec:ref-state-bcs. Note that this BC is only implemented with zero crossing.
(sec:bias-voltage-for-ac-and-cpp)=
If an AC potential boundary is coupled with a bias potential and an automatic adjustment of the AC amplitude to ensure
that a fixed power to the system is achieved, the BoundaryType has to be changed as compared with the previous section
BoundaryName = BC_left ! BC name in the mesh.h5 file
BoundaryType = (/52,1/) ! Dirichlet with 0V initial BC
where a RefState must be defined, which specifies the parameters for the sec:ref-state-bcs. Note that this BC is only implemented with zero crossing.
For details on the power coupling, see {ref}sec:fixed-coupled-power.
Dielectric material properties can be considered by defining regions (or specific elements) in the computational domain, where permittivity and permeability constants for linear isotropic non-lossy dielectrics are used. The interfaces between dielectrics and vacuum regions must be separated by element-element interfaces due to the DGSEM (Maxwell) and HDG (Poisson) solver requirements, but can vary spatially within these elements.
The dielectric module is activated by setting
DoDielectric = T
and specifying values for the permittivity
DielectricEpsR = X
DielectricMuR = X
Furthermore, the corresponding regions in which the dielectric materials are found must be defined, e.g., simple boxes via
xyzDielectricMinMax = (/0.0 , 1.0 , 0.0 , 1.0 , 0.0 , 1.0/)
for the actual dielectric region (vector with 6 entries yielding
xyzPhysicalMinMaxDielectric = (/0.0 , 1.0 , 0.0 , 1.0 , 0.0 , 1.0/)
Spherical regions can be defined by setting a radius value
DielectricRadiusValue = X
and special pre-defined regions (which also consider spatially varying material properties) may also be used, e.g.,
DielectricTestCase = FishEyeLens
where the following pre-defined cases are available as given in table {numref}tab:dielectric_test_cases
---
name: tab:dielectric_test_cases
---
| Option | Additional Parameters | Notes |
| :--------------------------: | :-----------------------: | :------------------------------------------------------------------------------------: |
| `FishEyeLens` | none | function with radial dependence: $\varepsilon_{r}=n_{0}^{2}/(1 + (r/r_{max})^{2})^{2}$ |
| `Circle` | `DielectricRadiusValue`, | Circular dielectric in x-y-direction (constant in z-direction) |
| | `DielectricRadiusValueB`, | with optional cut-out radius DielectricRadiusValueB along the |
| | `DielectricCircleAxis` | axis given by DielectricCircleAxis |
| `DielectricResonatorAntenna` | `DielectricRadiusValue` | Circular dielectric in x-y-direction (only elements with $z>0$) |
| `FH_lens` | none | specific geometry (`SUBROUTINE SetGeometry` yields more information) |
For the Maxwell solver (DGSEM), the interface fluxes between vacuum and dielectric regions can either be conserving or non-conserving, which is selected by
DielectricFluxNonConserving = T
which uses non-conserving fluxes. This is recommended for improved simulation results, as described in {cite}Copplestone2019b.
When particles are to be considered in a simulation, these are generally removed from dielectric
materials during the emission (inserting) stage, but may be allowed to exist within dielectrics by
setting
DielectricNoParticles = F
which is set true by default, hence, removing the particles.
Regions or zones (corresponding to zones as defined by hopr) can also be used to define dielectrics. In this case, the number of zones must be supplied
DielectricNbrOfZones = 8
and a vector of the same size for the corresponding zone IDs,
DielectricZoneID = (/9 , 10 , 11 , 12 , 13 , 14 , 15 , 16/)
DielectricZoneEpsR = (/2.0 , 2.0 , 2.0 , 2.0 , 2.0 , 2.0 , 2.0 , 2.0/)
DielectricZoneMuR = (/1.0 , 1.0 , 1.0 , 1.0 , 1.0 , 1.0 , 1.0 , 1.0/)
Note that this method cannot be used in combination with any other way of defining dielectrics.