Notes on support of partial differential equations (PDE) in VHDL‐AMS One topic of the 2013‐08‐13 IEEE P1076.1 WG meeting was a discussion on requirements for PDE support in VHDL‐AMS. At the end of the meeting, it was stimulated to collect more material for the further discussion. Some aspects regarding PDE support for system simulation tasks are presented in the following non‐exhaustive from the author’s personal point of view. 1 Areas of application Modeling microelectromechanical and nanoelectromechanical systems (MEMS and NEMS) Pelesko and Bernstein give an overview on modeling requirements based on partial differential equations for MEMS and NEMS [1]. Starting point for the development of models is in general the description of the relationship between continuum mechanics, heat conduction, fluid dynamics and electrostatics. The mathematical descriptions base on the subsequent equations and their simplifications for special application scenarios using 
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Navier equations for linear elasticity (see [1], eq. (2.41)] especially applied to o thin elastic membranes (see [1], eq. (2.49)) o (one‐dimensional) elastic beams with small thickness o (two‐dimensional) elastic plates (see [1], eq. (2.98)) Heat equation for linear thermoelasticity (see [1], eq. (2.9)) especially o modified w.r.t. the conversion of elastic energy into thermal energy (see [1], eq. (2.100)) (Nonlinear) Navier‐Stokes equations (see [1], eq. (2.120) and (2.121)) applied to o incompressible fluids (see [1], eq. (2.122) and (2.123) o and incompressible and inviscid fluids (described by the Euler equations, [1], eq. (2.124) and (2.125)) Maxwell equations for electrostatics (see [1], eq. (2.162) and (2.163)) o where a parabolic problem has to be solved by investigation the coupling between the deflection of a beam or membrane and the electrostatic force (see [2] and the discussion on pull‐in behavior) Simplifications are done regarding special geometries and considering only coupling between primary effects. This leads to lumped element models that may derived analytically (see for instance [1, 3, 4]) or by model order reduction (see for instance [5]). Prerequisite of this approach is in general a linear description of the model. Electromagnetic problems – rotational machines A special problem is the circuit coupling to rotating magnetic fields in rotational machines. 2D‐Poisson equation can be used to describe low frequency electromagnetic phenomena in rotating machines [6]. 2D magnetdynamic representation of cylindrical electrical machines is widely used (see for instance [7]) where the interaction to current excitation, mechanical loads and thermal coupling is also considered. 1 Compact Modeling Problems PDEs can be used to describe special problems of semiconductor device behavior coupled with DAE descriptions of the surrounding circuit. An overview on some mathematical problems is given in [8]. 2 PDE Support in modeling languages Modelica There were discussions on how to extend Modelica by statements for the description of PDE problems. This covers [9] 
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domain classes to define where an equation is hold specification of equations for a domain description of initial and boundary conditions This results in a proposal for PDEModelica [10]. Cellier and Dshabarow pursued a different approach [11] and prepared a PDELib package written in Modelica for PDE support [12]. At the present time, both approaches are not part of the Modelica standard. Matlab A PDE toolbox written in Matlab was created to solve PDE problems within Matlab [13]. It supports processing of 2D PDEs  elliptic PDEs  parabolic, hyperbolic, and Eigenvalue PDEs  nonlinear elliptic PDEs Verilog‐A The Verilog‐AMS LRM provides a ddx operator that delivers the derivative of an expression w.r.t. a variable other than time (see section 4.5.4 of [14]). This feature can be applied for compact modeling problems [15]. 3 Coupling There are several solutions known that support a co‐simulation of system simulation tools and tools that support PDE simulation. Examples are for instance  seamless integration of COMSOL Multiphysics with Matlab [16]  ANSYS Maxwell and Simplorer [17] 2 4 Summary Different approaches for PDE handling in VHDL‐AMS are possible: 
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Extension of the VHDL‐AMS by new language statements. A small extension would be to provide an operator that provides the derivative to a variable other than time (similar to Verilog‐A). Extensive extensions would include the formulation of partial differential equations within VHDL‐AMS (similar to PDEModelica). Description of at least 2D problems requires a significant effort. The application area for 1D solutions seems to be limited. Preparation of a package for special classes of equations (similar to PDELib for Modelica or the Matlab PDE toolbox) Supply of a standardized interface for the coupling to specialized PDE tools. From the users’ point of view, this approach would help to include existing component models based on PDE into the system simulation. References [1] John A. Pelesko and David H. Bernstein, Modeling MEMS and NEMS, CRC Press, 2003. Online: http://books.google.de/books?id=_jCIat4JkwoC&printsec=frontcover&hl=de&source=gbs_ge_summary_r&cad=0#v=onepage&q&f=true [2] Pierpaolo Esposito, Nassif Ghoussoub and Yujin Guo, Mathematical Analysis of Partial DiffrerentialEquations Modeling Electrostatic MEMS, American Mathematical Society, 2010. Online: http://books.google.de/books?id=mljLAmcaT6kC&printsec=frontcover&hl=de&source=gbs_ge_summary_r&cad=0#v=onepage&q&f=true [3] Sergey Edward Lyhevski, Nano‐ and Micro‐Electromechanical Systems, CRC Press, 2006. Link: http://www.crcpress.com/product/isbn/9780849328381 [4] Stephen D. Senturia, Microsystem Design, Springer, 2001. Link: http://www.springer.com/engineering/electronics/book/978‐0‐7923‐7246‐2 [5] E. B. Rudnyi and J. G. Korvink, Model Order reduction of MEMS for Efficient Computer Aided Design and System Simulation, Lecture Notes in Computer Science, v. 3732, pp. 349‐356, 2006. Online: http://modelreduction.com/doc/papers/rudnyi04MTNS.pdf [6] Salon, S.J, Finite Element Analysis of Electrical Machinery. Online: http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=53227&userType=inst [7] De Gersem, H.; Hameyer, K.; Weiland, T., Skew interface conditions in 2‐D finite‐element machine models, IEEE Transactions on Magnetics, vol.39, no.3, pp.1452,1455, May 2003 Online: http://dx.doi.org/10.1109/TMAG.2003.810543 [8] Caren Tischendorf, Coupled Systems of Differential Algebraic and Partial Differential Equations in Circuit and Device Simulation, Habilitation, Humboldt Universität Berlin, 2003. Online: http://www.mathematik.hu‐berlin.de/~caren/pub/habilitation.pdf [9] Levon Saldamli and Peter Fritzson, Object‐oriented Modeling with Partial Differential Equations, Proc. Modelica Workshop, 2000. Online: https://modelica.org/events/workshop2000/proceedings/Saldamli.pdf 3 [10] Levon Saldamli, PDEModelica ‐ A High‐Level Language for Modeling with Partial Differential Equations, Dissertation, Linköping, 2006. Online: http://liu.diva‐portal.org/smash/get/diva2:22306/FULLTEXT01 [11] Farid Dshabarow and François E. Cellier, Dirk Zimmer, Support for Dymola in the Modeling and Simulation of Physical Systems with Distributed Parameters, Modelica 2008 Conference. Online: https://www.modelica.org/events/modelica2008/Proceedings/sessions/session6b3.pdf [12] PDELib Online: https://github.com/modelica‐3rdparty/PDELib [13] Partial Differential Equation Toolbox, Mathworks Online: http://www.mathworks.de/products/pde/ [14] Verilog‐AMS Language Reference Manual, Version 2.3.1 ‐ June 1, 2009. Accelera. Online: http://www.verilog.org/verilog‐ams/htmlpages/public‐docs/lrm/2.3.1/VAMS‐LRM‐2‐3‐1.pdf [15] Geoffrey J. Coram, How to (and how not to) write a compact model in Verilog‐A. Proc. BMAS 2004. Online: http://www.bmas‐conf.org/2004/papers/bmas04‐coram.pdf [16] COMSOL Website: LiveLink for Matlab. Online: http://www.comsol.com/products/livelink‐matlab/ [17] ANSYS Website: Dynamic Link with ANSYS Simplorer. Online: http://www.ansys.com/Products/Simulation+Technology/Electromagnetics/Electromechanical+&+Power+E
lectronics+&+Mechatronics/ANSYS+Maxwell/Features/Maxwell+‐+Dynamic+Link+with+Simplorer Contact: Joachim Haase, E‐Mail: joachim.haase@eas.iis.fraunhofer.de 2013‐09‐11 4