Danie Ludick MScEng Study Leader: Prof. D.B. Davidson Computational Electromagnetics Group Stellenbosch University Extended Studies of Focal Plane Arrays for the SKA and the MeerKAT

Focal Plane Array Feed-structures for the SKA Parabolic Dish Reflector Antennas for the SKA Typical Radio Astronomy Receivers Typical Radio Astronomy Receivers MeerKAT – already built a prototype MeerKAT – already built a prototype - the XDM (15 m dish) - the XDM (15 m dish) Requires a feed-structure at the focal point Requires a feed-structure at the focal point to collect the focused energy Focal Plane Arrays Focal Plane Arrays Some Applications of SKA requires whole sky Some Applications of SKA requires whole sky imaging – very time consuming with conventional single pixel feeds, such as horn antennas Solution: Reduce survey time by using focal plane Solution: Reduce survey time by using focal plane arrays → multiple beams can be formed that can be scanned independently of each other Focal Point XDM

Focal Plane Array Design Difficulty … Simulating the FPA Structure Large FPA designs, such as an array Large FPA designs, such as an array of Vivaldi Antennas → memory intensive, simulations can take several hours even days ! Need to investigate effective simulation techniques based on the Method of Moments Using FEKO in a High Performance Parallel Computing Environment Developing efficient solution techniques using the Characteristic Basis Function Method (CBFM)

The Method of Moments The Method of Moments (MoM) … Computational EM method for determining unknown surface current distribution on an electromagnetic scatterer, when illuminated by various excitations Computational EM method for determining unknown surface current distribution on an electromagnetic scatterer, when illuminated by various excitations Computational Cost of ~ O(N 2 ) memory storage and ~ O(N 3 ) for solving linear equations [A]{x} = {y} (size N x N) N = Unknowns, i.e. current associated with non-boundary edges of triangles Discritise the problem Solve for {x}

Using FEKO in a High Performance Computing (HPC) environment The Benchmarking Study … Simulating various sized Vivaldi Arrays on a Cluster at the CHPC in Cape Town Simulating various sized Vivaldi Arrays on a Cluster at the CHPC in Cape Town CHPC Infrastructure … IBM e1350 Linux Cluster … the iQudu 160 Compute Nodes → 2 dual-core AMD Opteron 2.6 GHz Processors Compute Nodes → 2 dual-core AMD Opteron 2.6 GHz Processors + 16 GByte of RAM per Node In Total 640 Processors In Total 640 Processors ~ 2.5 Teraflops Processing Power ~ 2.5 Teraflops Processing Power 2 Available Interconnects … 2 Available Interconnects … 1 GByte Ethernet and 10 GBit Infiniband Number of elements in array Number of Unknowns (N) 5504,3688,73617,47233,62438,34063,400 the iQudu

Example of HPC FPA Simulation 64 Element Vivaldi Array 32 Nodes 32 Nodes 33,624 Unknowns 33,624 Unknowns 25 Frequency Points Analyzed 25 Frequency Points Analyzed (400 MHz – 2 GHz) Results On 1 node, estimated runtime is ~ 40 hours (1.6 days) ! SimulationPrediction Total Runtime ~ 4 hours ~ 4.5 hours Memory Usage GByte 8.4 GByte |S 11 | [dB] Frequency [GHz]

What we have in terms of simulation power Fast … but not fast enough

What we want … To achieve this … improve algorithms

The Characteristic Basis Function Method (CBFM) Conventional Solvers … Direct Methods Direct Methods (Fast, N is limited) Iterative Solvers Iterative Solvers (Big problems, very long runtimes) The CBFM … Reduces the size of the matrix equations Reduces the size of the matrix equations Solve using direct methods Solve using direct methods [A]{x} = {y} (size N x N) MoM Solve

The Characteristic Basis Function Method (CBFM) … introductory example Z X Y 3λ3λ 3λ3λ N i unknowns ([A]{x} = {y}) NxN N ~ 1000’s 2 HE Z (1)Sub-domains (1 … 4) (2)Primary CBFs, J i (i) i = (3)Secondary CBFs, J k (i) i = 1..4; k = 1.. i-1, i+1,.. 4 (4)Entire solution to the problem … (5) Reduced [A]{x} = {y} (6) Solve this equation for the complex α coefficients with direct methods α coefficients with direct methods                 )( 1 )( 1 )2( )2( 1 )1( )1( 1 : : 0 0 : 0 M k M k M k M k k k M k k k N J J J x 

The Characteristic Basis Function Method (CBFM) – Numerical Results Results show Current Distribution on 4 x 1.5λ plates (λ/12 discritization) Number of Unknowns (N) = 3,744 Number of CBFM Sub-domains (M) = 4 Number of CBFs (M 2 ) = 16 Runtime reduces by factor of ~ 13 Iterative Methods CBFM Iterative Methods CBFM MethodCGSGMRESCBFM Solution Time (s) Iterative Solutions include Preconditioning

The Way Forward HPC resources (e.g. the iQudu) and methods such as CBFM provides basis for simulating very large Focal Plane Arrays CBFM is highly parallelizable … implement on cluster such as iQudu … dramatically reduce simulation time Apply efficient numerical techniques to various FPA’s (Vivaldi, Checkerboard, New Ideas …) Goal: Provide SKA Engineers with a means of analyzing large FPA feed-structures efficiently

Questions ?