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Application of neBEM to solve MPGD electrostatics Supratik Mukhopadhyay, Nayana Majumdar, Sudeb Bhattacharya Saha Institute of Nuclear Physics, Kolkata, India Presented by Rob Veenhof CERN

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Analytic FEM / FDM Nearly arbitrary geometry Flexible Exact Simple interpretation BEM Reduced dimension Accurate for both potential and its gradient Solve Poisson’s equation x Complex numerics x Numerical boundary layer x Numerical and physical singularities x Interpolation for non- nodal points x Numerical differentiation for field gradient x Difficulty in unbounded domains x Restricted x 2D geometry x Small set of geometries Field Solver

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Finite Element Blues The complete 3D volume is discretized using nodes distributed throughout the volume The governing equation is satisfied at the nodal points to solve for potential The variation of potential from node to node is determined by a basis function – usually a low order polynomial Fields are thus represented by even lower order polynomial Value of potential at an arbitrary location is obtained by interpolating values from surrounding nodes – inaccurate At a non-nodal location, field values suffer even more Serious problems in near-field Artificial truncation of far-field boundary is a necessity for problems with open domain Mesh used for micromegas FEM solution by P Cwetansky

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BEM Advantages Solves for charge density distribution on each of the boundary elements by adopting a Green function approach Once the charge density distribution is estimated correctly, potential and flux can be easily obtained at any arbitrary location – essentially mesh-less No precision-eroding interpolation and extrapolation are ever necessary Near-field can be seamlessly handled if charge density distribution is appropriate Far-field conditions are satisfied naturally for open-domain problems Discretization is easier due to reduction of dimensionality of the problem. It is also easier to discretize complex devices. Mesh used for micromegas neBEM solution Anode strip Micromegas Drift plane

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BEM Basics Potential u at any point y in the domain V enclosed by a surface S is given by where y is in V, u is the potential function, q = u,n, the normal derivative of u on the boundary, b(x) is the body source, y is the load point and x, the field point. U and Q are fundamental solutions U 2D = (1/2 ) ln(r), U 3D = 1 / (4 r), Q = -(1/2 r ) r,n = 1 for 2D and 2 for 3D. Distance from y to x is r, n i denotes the components of the outward normal vector of the boundary. Green’s identitiesBoundary Integral Equations 2D Case3D Caser = 0r 0, r≠ 0 ln(r)1/rWeak singularityNearly weak singularity 1/r1/r 2 Strong singularityNearly strong singularity 1/r 2 1/r 3 Hyper singularityNearly hyper-singularity

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Solution of 3D Poisson's Equation using BEM Numerical implementation of boundary integral equations (BIE) based on Green’s function by discretization of boundary. Boundary elements endowed with distribution of sources, doublets, dipoles, vortices. Electrostatics BIE Charge density at r’ Green’s function - permittivity of medium discretization Accuracy depends critically on the estimation of [A], in turn, the integration of G, which involves singularities when r → r'. Most BEM solvers fail here. Potential at r Influence Coefficient Matrix { ρ } = [A] -1 { Φ}

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While computing the influences of the singularities, the singularities are modelled by a sum of known basis functions with constant unknown coefficients. The strengths of the singularities are solved depending upon the boundary conditions, modeled by shape functions. Singularities are assumed to be concentrated at centroids of elements (thus avoiding integration), except for cases such as self influence. r = r’: Mathematical singularities can be removed; Sufficient to satisfy the boundary conditions at centroids of the elements. Numerical boundary layer r -> r’: Difficulties in modeling physical singularities geometric singularity: Closely spaced surfaces, corners, edges boundary condition singularity: Dirichlet and Neumann conditions close-by Conventional BEM

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Present Approach Foundation expressions are analytic and valid for the complete physical domain Analytic expressions of the integration yielding both potential and flux field at any arbitrary location due to a uniform distribution of source on flat rectangular and triangular elements have been derived using symbolic tools. Using these elements, surfaces of any 3D geometry can be discretized without requiring the singularities to be concentrated only at certain specific nodes. Restatement of the approximations Singularities distributed uniformly on the surface of boundary elements. Nodal concentrations, rather than the integrations, are avoided altogether. Strength of the singularity changes from element to element (unchanged). Strengths of the singularities solved depending upon the boundary conditions, modeled by the shape functions (unchanged) ISLES library and neBEM Solver

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Contrast of approaches nodal (conventional) versus distributed (neBEM) Realistic representation: Accurate solutions everywhere, including near-field Aspect ratio of element sides can be much larger Size of elements can be varied easily Edges, corner, closely packed surfaces can be modeled easily Proximity of Dirichlet and Neumann conditions allowed Single formulation for many problems Unrealistic representation: Near-field solutions grossly incorrect Aspect ratio of element sides cannot be high Size of elements cannot be varied sharply Edges, corners, closely packed surfaces cannot be modeled easily Proximity of Dirichlet and Neumann condition cannot be allowed Flourish of special formulations!

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Precision in flux computation comparison with quadrature (nodal approach of ususal BEM) Quadrature with even the highest discretization fails! Quadrature with only the highest discretization produces results comparable to ISLES zMax = 10.0 Engineering Analysis with Boundary Elements (EABE), Elsevier, Available online 3 August 2008

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Precision in flux computation Comparison with multipole expansions The quadrupole results are still far from precise Comparison of flux along a line parallel to the Z axis passing through the barycenter Comparison of flux along a diagonal passing through the barycenter EABE, online, 3 Aug 2008

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Expected features of a Field Solver for MPGDs Variation of field over length scales of a micron to a meter needs to be precisely estimated Fields at arbitrary locations should be available on demand Intricate geometrical features – essential to use triangular elements, if needed Multiple dielectric devices Nearly degenerate surfaces Space charge effects can be very significant Dynamic charging processes may be important It may be necessary to calculate field for the same geometry, but with different electric configuration, repeatedly GEM Typical dimensions: Electrodes (5 μm thick) Insulator (50 μm thick) Hole size D ~ 60 μm Pitch p ~ 140 μm Induction gap: 1.0 mm, Transfer gap: 1.5 mm Micromegas Typical dimensions: Mesh size: 50 μm Micromesh sustained by 50 μm pillars

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Micro-wire: a test case Representative length variation Closely spaced elements / surfaces Multiple dielectric device Complex geometry Field variation very close to anode is important FEM solutions available (Peter Cwetansky: garfield/examples/micropattern/ microwire/index.htm) Please note that for computation, a similar, but not identical geometry has been used Source: Adeva et al, USC-FP/99-01

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Discretization In FEM, the complete 3D domain needs to be discretized Far-field in FEM is truncated and, possibly, supplied with Neumann boundary conditions to maintain periodicity and assign drift field strength In neBEM, the far- field is realistically represented by a drift plane at a certain voltage.

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Close up Elements of various sizes and aspect ratios have been used Since we are interested mostly in the field variation around the anode, it has been discretized using very small elements The lines join the element centroids – it is not the true mesh and, hence, has apparent gaps

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Influence coefficient matrix Consider a system of two conductors, each having been discretized into two elements. In order to generalize the situation, let us even consider one of the conductors to be floating. Thus, one of these conductors is at a known voltage, V. The other conductor is at a floating voltage V F, which is unknown. Number the elements on the conductor with known voltage to be 1, 2, and those on the floating conductor to be 3, 4. Denote charge densities by i, area by A i, on each element Resulting system of equation is as shown – the last equation reflecting the fact that the total charge on a floating conducting object is zero In the above system, I ij denotes the influence of the j th element on the i th element. Please note that if we have more than one floating conductor, they cannot be assumed to be at the same potential, and one column and one row as shown above needs to be added for each floating conductor. Please note that the matrix is not sparse.

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Evaluation of influence coefficients 2 terms 4+4 terms 2 terms 4 log terms 4+4 complex tanh -1 terms Similar, but more complex, expressions have been derived for trianglular elements

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Charge density distribution The neBEM solves for the charge density On the cathode, the density is negative and with relatively less variation On the kaptons, the density is of both polarities and the variation is least On the anode, the density is positive and the variation is very sharp, the edges and corners having very large charge densities It is at these edges and corners where the FEM fails, in general Anode Cathode Kapton

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Intricate geometries The Micro Wire Detector The MWD has an intricate design. In this case: Drift plane 785μm from the anode strip at 1.11kV. Total electric field contours on the central plane across cathode and anode Variation of total electric field along an axis passing through the mesh hole JINST, 2007, 2 P09006

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Case studies These are the possible design variation discussed in the earlier study, the conventional one being termed as mesh, other one as segmented. The latter consumes 40% less copper!

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Potential There is significant variation in the potential contours of these two possible designs (points representing `mesh’ configuration, broken lines for `segmented’). The right figure shows the potential for the segmented design only.

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Flux There is significant variation in the flux contours as well (points representing `mesh’ configuration, broken lines for `segmented’, once again). The right figure shows the potential for the segmented design only.

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Comparison with FEM The mesh and segmented configurations have significant differences except at the mid-zone of the detector unit (please note that the Electric field is on a log scale) Close to the anode, the mesh configuration achieves a higher field The drift field is more for the segmented configuration Since the far-field for the FEM computations is treated quite differently from the neBEM, values at these zones differ considerably (once again, note that the scale is logarithmic and the variation is, in fact, less than what is apparent Besides the far-field, the two results agree quite closely

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Comparison with FEM (near-field) Field around a line just 1μm away from the anode surface is considered here –sampling for neBEM is as small as 0.1μm! The mesh configuration has higher field values throughout Sharp rise in the field values is observed at all the four edges Smooth variation of field is observed on each of the four surfaces Field values are found to decrease sharply once the points are beyond anode surfaces FEM computation is clearly unable to produce correct results near and at the edges FEM, although better on the surfaces, still falls behind neBEM in performance

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Effect of discretization (near-field) In the earlier computation, we had used 20 elements to represent the top surface and 10 elements on the side surface. The elements were made successively smaller towards the edges In order to study effect of using coarse discretization, we also used larger elements of fixed size – only 3 elements each to represent both top and side surfaces Although there is significant difference between the results, the overall trend is represented well by the larger elements It is important to note that there is no jaggedness (at 0.1μm sampling) despite the use of unreasonably large elements!

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Multiple dielectric devices Resistive plate chambers Strip width: 3.0cm, Strip length: 50.0cm Layer height: 2.0mm Layer-3 permittivity ( r ): 7.75 (~glass) Layer-2 (middle) permittivity ( r ): (~Argon) Successful validation with Riegler et al. Layer-4,5 height: 200µm (~PET) Layer-6,7 height: 20µm (~Graphite) Layer-4,5 permittivity ( r ): 3.0 (~PET) Layer-6,7 permittivity ( r ): 12.0 (~Graphite) NIM, A 595 (2008)

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Conclusions We have outlined the approach neBEM takes in order to compute the field properties in a given device The approach has been compared with the more standard FEM method in fair detail The micro-wire detector has been analyzed since it has features typical of many of the MPGDs, and has good FEM results available for it FEM and neBEM results have been compared for far- and near-field neBEM results have been found to be more accurate in critical regions Large variation of discretization has not deteriorated neBEM results to any great extent RPC weighting field results have been presented in order to demonstrate the accuracy of neBEM – comparisons against analytical results have turned out to be encouraging

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Plans for 2009 Development of an interface to ROOT so that devices built using ROOT can be directly imported to neBEM and solved for For simple shapes, thanks to Andrei, we have already been able to extract the surfaces that can be exported to neBEM using a ROOT script or a stand-alone C++ code For composite shapes, thanks to Timur, we have been able to get the elements being used for the geometrical rendering in ROOT. We are trying to understand this mechanism to be finally able to export these surfaces in the neBEM format A working interface to garfield and the new detailed detector simulation framework is expected to be complete by the middle of 2009 We will also try to set up an experimental laboratory for the development of MPGDs during this period

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Plans for beyond 2009 oThe problem of dynamic charging will be addressed. oParticles on Surface (ParSue), a new model for space-charge simulation based on this formulation has been proposed recently. This model needs to be explored and integrated properly. oProblems related to magnetostatics will be addressed. oOptimization and introduction of adaptivity in the process of mesh generation will be implemented oImplementation of improved and more efficient matrix solution algorithms will be carried out. oParallel computation can help the overall detailed simulation and efforts may be made in this direction oA toolkit version of the field-solver may be developed for use in other areas governed by the Poisson’s equation.

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Acknowledgements We thank the organizers for giving us a chance to present this work We thank our Director, Prof. Bikash Sinha, for his support We thank Rob for encouraging us, suggesting improvements and test cases for neBEM and, finally, for presenting this material We thank you all for your kind attention Looking forward to a very friendly and successful collaboration !

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