D3D VIEQ A Quantum Leap

D3D Volume Integral Equation tackles ultra-large electromagnetic problems at a scale simply out of reach for any other solver.

D3D VIEQ solves full-wave, real-world geometries with millions of elements on engineering desktop workstations.

  • Only discretizes parts of the model featuring dielectric or magnetic contrast
  • Utilizes tetrahedral elements highly economically
  • Requires no discretization of air or vacuum space surrounding the structure
  • Eliminates the need for absorbing boundary conditions of any kind
  • Completely avoids boundary truncation errors
  • Implements the approximate convolution method, entirely removing the scalability barrier

D3D The User Interface Revolution

computational electrodynamics - dr. benedikt oswald introduces a revolutionary, AI-assisted, agentic browser interface for seamless cloud access to the D3D codes.

Engineers provide design specifications via:

  • Text prompts
  • Hand-drawn sketches
  • PDF documents
  • Complete CAD geometries

The interface autonomously:

  1. Analyzes your input and creates your electromagnetic project.
  2. Executes your project using D3D solvers.
  3. Analyzes and presents essential results to you.
  4. Iterates based on your feedback until design specifications are met.

Your data confidentiality is paramount. We rely exclusively on secure and trustworthy AI models.

Solver Convergence
Solver Convergence
Computational Mesh
Computational Mesh
Computed Electric Field
Computed Electric Field
Skywalker Model
Skywalker Model
Computed Electric Field 1 GHz
Computed Electric Field 1 GHz
Solver Convergence
Solver Convergence
Computational Mesh
Computational Mesh
Computed Electric Field
Computed Electric Field
Skywalker Model
Skywalker Model
Computed Electric Field 1 GHz
Computed Electric Field 1 GHz

ELECTROMAGNETIC MANIFESTO - from the molecules of life to the latest communications and measurement technologies our world is built and depends on electromagnetic fields.

For the prosperous future intuitive understanding of all phenomena electrodynamic is vital. Engineers and scientists need new, intuitive tools.

The D3D solver collection provides these tools.

Technical Index

COMPUTATIONAL ELECTRODYNAMICS  /  FINITE ELEMENT METHOD  /  DISCONTINUOUS GALERKIN  /  INTEGRAL EQUATION METHODS  /  FREQUENCY DOMAIN  /  TIME DOMAIN  /  EIGENMODE ANALYSIS  /  APPROXIMATE CONVOLUTION METHOD  /  AI ASSISTED, INTUITIVE MESH GENERATION  /  BASED ON SOLID MODELING KERNEL  /  DISTRIBUTED MEMORY SPARSE LINEAR SOLVER  /  ITERATIVE KRYLOV SUBSPACE LINEAR SOLVER  /  GEOMETRY OPTIMIZATION  /  BASED ON SOLID MODELING KERNEL  /  ITERATIVE SOLVER

D3D VIEQ

D3D Volume Integral Equation code solves the electric field volume integral equation (VIEQ) in the frequency domain.

Specification Datasheet

D3D VIEQ Technical Capabilities

Code Feature Implementation Dimensionality Status
AI Assisted Frontend conversational electromagnetic project setup 3-dimensional upcoming release
Electromagnetics electric field volume integral equation 3-dimensional operational
Mathematics discontinuous Galerkin finite element 3-dimensional operational
Element type unstructured tetrahedral mesh 3-dimensional operational
Element order 1st order, i.e. straight edge tetrahedral 3-dimensional operational
Element order 2nd order, i.e. quadratic curvilinear tetrahedral 3-dimensional operational
Basis function hierarchical tangential vector finite element (HTVFE) 3-dimensional operational
Basis function order 1st order 3-dimensional operational
Basis function order 2nd order 3-dimensional operational
Basis function order 3rd order 3-dimensional upcoming release
Dielectric materials constant complex relative permittivity — operational
Dielectric materials measured frequency dependent complex relative permittivity — operational
Magnetic materials constant complex relative permeability — operational
Magnetic materials measured, frequency dependent complex relative permeability — operational
Conducting materials real valued ohmic conductivity — operational
Periodicity Periodic Boundary Conditions (PBC) 2-dimensional operational
Sheet Material perfect electric conductor (PEC) — operational
Sheet Material perfect magnetic conductor (PMC) — operational
Source transverse electric magnetic (TEM) wave linear polarization operational
Source Hertzian dipole 3-dimensional operational
Source Waveguide ports — operational
Source transverse electric magnetic (TEM) wave. circular & elliptical polarization — operational
Parallelization shared memory parallelization via OpenMP — operational
Parallelization distributed memory parallelization via MPI — operational
GPU Acceleration Apple Metal — operational
GPU Acceleration NVIDIA — upcoming release
Linear System Solver BICGSTAB — operational
Linear System Solver GMRES — operational
Linear System Solver GCRODR — operational
Preconditioner Block Jacobi — operational
Preconditioner Overlapping Block Jacobi — operational
Preconditioner ILU(0) — operational
Preconditioner Gauss Seidel — operational
Preconditioner SPAI — operational
Preconditioner SSOR — operational
Electric field complete field 3-dimensional operational
Magnetic near field complete field 3-dimensional operational
Scattered electric field selectable resolution 3-dimensional operational
Scattered magnetic field selectable resolution 3-dimensional operational
Electric far field selectable resolution 3-dimensional operational
Magnetic far field selectable resolution 3-dimensional operational
Radar Cross Section (RCS) selectable resolution 3-dimensional operational
Scattering coefficients via waveguide port over frequency operational
Extinction cross section via near field 3-dimensional operational
Absorption cross section via near field 3-dimensional operational
Radiation cross section via far field 3-dimensional operational
Showing 46 of 46 capabilities D3D VIEQ Specification Datasheet • 17. September 2026

D3D DGFD

D3D Discontinuous Galerkin Frequency Domain solves the electric field vector wave equation in the frequency domain with a discontinuous Galerkin method. The D3D DGFD code is currently tested and will be accessible for customers in late 2026, maybe earlier for selected customers. D3D DGFD solves the toughest electrodynamic problems, is parallelized for shared memory server boards and distributed memory cluster computers. D3D DGFD scales from the modest laptop to the ultra-powerful supercomputer.

D3D CGTD

D3D Continuous Galerkin Time Domain solves the electric field vector wave equation in the time domain with a continuous Galerkin method. The D3D CGTD code is currently tested and will be accessible for customers in late 2026, maybe earlier for selected customers. D3D CGTD solves the largest electrodynamic problems, models dispersive dielectric materials, is parallelized for shared memory server boards and distributed memory cluster super computers. D3D CGTD scales from the modest laptop to the ultra-powerful supercomputer.

Demokritos3D Benchmark Suite

Benchmarks

Computational precision across physical application domains. In accordance with our principles, detailed benchmark reports—including quantitative accuracy figures, convergence history, and solver telemetry—are distributed directly to known partners upon request.

Scenario 01  |  Microwave & Millimeter Wave

1. Microwave & Millimeter Wave Benchmarks

High-frequency full-wave volume and surface solutions. Accurately predicts high-contrast dielectric resonators, dielectric resonator antennas (DRA), heterogeneous biological tissues, and canonical spherical scatterers.

Target Geometries
Dielectric Resonators & Scatterers

Canonical dielectric spheres, dielectric resonator antennas (DRA), and inhomogeneous biological tissue phantoms.

VIEQ • Mixed VSIE
Analytical Baseline
Mie Scattering & RCS Far-Field

Cross-validated against exact analytical Mie series across observation directions. In addition to near-fields, we offer full Radar Cross Section (RCS) far-field radiation and scattering plots, confirming strict energy conservation and reciprocity.

Exact Mie Series • RCS Far-Field Plots • Strict Reciprocity
Discretization & Solver
Polynomial Tetrahedra + Approximate Convolution

High-order polynomial volumetric basis functions on tetrahedra. The dense direct solver is used as an exact reference whenever problem dimensions permit, while iterative solver schemes accelerated by approximate convolution are deployed on a routine basis to solve ultra-large electromagnetic problems.

Dense Reference • Routine Iterative • Ultra-Large Problems
Dossier Contents
Full Benchmark Dossier

Complete execution logs, condition numbers, convergence metrics, Radar Cross Section (RCS) far-field plots, near-field slices, and analytical comparison datasets are compiled in the official benchmark document.

RCS Far-Field • Verified Telemetry • Technical PDF
Direct Technical Inquiry

Request Microwave & MMW Suite

Detailed benchmark documents, verification data, and raw solver telemetry are provided directly to qualified engineering and research partners.

REQUEST BENCHMARK SUITE →
Handled personally by Dr. Benedikt Oswald
Scenario 02  |  Monostatic & Bistatic Radar Signatures

2. Radar Cross Section (RCS) Benchmarks

Electromagnetic signature predictions validated against experimental anechoic chamber measurements from the University of Texas at Austin suite and aerospace airframe platforms.

Target Geometries
NASA Almond Benchmark Suite

NASA Almond benchmark models across physical configurations: closed PEC, solid resin, closed tail-coated, and open tail-coated structures.

PEC • Dielectric • Mixed VSIE
Experimental Validation
UT Austin Anechoic Chamber

Monostatic and bistatic azimuth sweeps validated against experimental anechoic chamber measurements and independent computational reference solutions for both polarizations.

Anechoic Measurement Match • VV / HH Polarizations
Iterative Acceleration
Krylov Subspace + Approximate Convolution

High-capacity volumetric iterative solvers accelerated by approximate convolution, capturing tip diffraction, surface creeping waves, and coated cavity resonances.

Fast Iterative Solver • Multi-Core OpenMP
Dossier Contents
Full Benchmark Dossier

Polar and Cartesian radar cross-section plots, experimental measurement overlays, and iteration convergence data are compiled in the official benchmark document.

Full Polarization Data • Technical PDF
Direct Technical Inquiry

Request RCS Benchmark Suite

Detailed benchmark documents, verification data, and raw solver telemetry are provided directly to qualified industrial and defense partners.

REQUEST BENCHMARK SUITE →
Handled personally by Dr. Benedikt Oswald
Scenario 03  |  Visible & Near-Infrared Spectrum

3. Optical & Plasmonic Benchmarks

Sub-wavelength nanophotonics and localized surface plasmon resonance (LSPR). Verified against exact analytical Mie series with dispersive experimental metal permittivities.

Target Geometries
Metallic & Dielectric Nanostructures

Gold and silver nanosphere resonators, dimers, nanorod antennas, and high-index dielectric nano-resonators.

LSPR • Dispersive Permittivities
Analytical Baseline
Exact Analytical Mie Series

Cross-checked against exact analytical Mie theory across the resonance spectrum for extinction, scattering, and absorption cross-sections.

Exact Mie Benchmark • Full Spectral Sweeps
Discretization & Skin Depth
High-Order Tetrahedral Meshing

High-order polynomial volumetric basis functions resolving sub-wavelength skin-depth gradients and boundary-layer near-field singularities in metallic nanoparticles.

High-Order Tetrahedra • Boundary Layer Meshing
Dossier Contents
Full Benchmark Dossier

Extinction and scattering spectra, Poynting absorption balances, near-field electric field vector maps, and solver execution timings are compiled in the official benchmark document.

Spectral Sweeps • Vector Field Slices
Direct Technical Inquiry

Request Optical Suite

Detailed benchmark documents, verification data, and raw solver telemetry are provided directly to qualified nanophotonics and plasmonics research teams.

REQUEST BENCHMARK SUITE →
Handled personally by Dr. Benedikt Oswald
Scenario 04  |  Extreme Ultraviolet (EUV)

4. Extreme Ultraviolet (EUV) Benchmarks

Computational electrodynamics for sub-nanometer extreme ultraviolet lithography regimes. Full-wave 3D electromagnetic modeling of near-field diffraction and material interactions.

Extreme Ultraviolet (EUV) Application Examples Currently Prepared

Part of the work for the upcoming solver release. Extreme ultraviolet computational electrodynamics benchmarks and verification datasets are currently in active preparation.

Development Scope
3D EUV Computational Modeling

Full-wave computational electrodynamics for extreme ultraviolet regimes, addressing 3D diffraction and near-field interactions at the sub-nanometer scale.

Sub-Nanometer Regime • 3D Diffraction
Solver Approach
Volume Integral Maxwell Formulation

Rigorous volume integral formulation engineered to capture volumetric optical contrast, phase variation, and geometric details without artificial boundary truncation.

Volume Integral • True 3D Scattering
Verification Principles
Rigorous Benchmark Standards

In alignment with our engineering standards, computational benchmarks undergo extensive multi-level verification and sanity checks before public release.

Multi-Level Verification • Rigorous Standards
Upcoming Release
Benchmarking In Preparation

Benchmarking and telemetry datasets are actively being prepared for the upcoming solver release. Collaborative evaluation inquiries are welcome.

Upcoming Release • Collaborative Evaluation
Pre-Release Technical Inquiry

Contact Us Directly About The EUV Benchmark Suite

Discuss extreme ultraviolet computational electrodynamics capabilities and pre-release evaluation directly with our technical team.

CONTACT US DIRECTLY ABOUT THE EUV BENCHMARK SUITE →
Handled personally by Dr. Benedikt Oswald
Permanent Ledger

News Archive

A chronological record of our continuous development and operational progress. We overachieve.