A Novel Mesh Quality Improvement Method for Boundary Elements

Algorithm 1

Nonsmooth optimization algorithm.

(1)  function SOLUTION (𝑓(𝑥), 𝜆₀, 𝜀, 𝑋⁰_𝑛, 𝑁)

     ▸  𝑓(𝑥) = objective function

     𝜆₀ = initial step size

     𝜀= desired degree of accuracy

     𝑁= iteration number

     𝑋⁰_𝑛= initial coordinate for vertices X

(2) get free vertex 𝐻⁰_𝑛𝑎𝑛𝑑𝑘=0

  ▸  𝐻⁰_𝑛 = Hessian matrix

  𝑘= Iteration number

(3) min 𝑓(𝑋^𝑘_𝑛+𝜆_𝑘𝑑^𝑘_𝑛)

  ▸  𝑑^𝑘_𝑛= optimal search direction

(4) compute 𝜆^∗_𝑘 using line search algorithm (see Algorithm 2)

  ▸  𝜆^∗_𝑘 = optimal step size

(5) 𝑋_𝑛^𝑘+1=𝑋^𝑘_𝑛+𝜆^∗_𝑘𝑑^𝑘_𝑛

(6)  if  |𝑓(𝑋_𝑛^𝑘+1)−𝑓(𝑋^𝑘_𝑛)|<𝜀 or 𝑘>𝑁 then

(7) 𝑋^∗_𝑛←𝑋_𝑛^𝑘+1 and 𝐛𝐫𝐞𝐚𝐤

  ▸  𝑋^∗_𝑛= optimal point coordinate

(8) else Δ𝑋_𝑛^𝑘+1=𝑋_𝑛^𝑘+1−𝑋_𝑛^𝑘+1,Δ𝑔^𝑘_𝑛=𝑔_𝑛^𝑘+1−𝑔^𝑘_𝑛

  ▸  𝑔_𝑛^𝑘+1=𝑓(𝑥) gradient at vertices 𝑋_𝑛^𝑘+1

  𝐻_𝑛^𝑘+1=(𝐻^𝑘_𝑛+Δ𝑋^𝑘_𝑛(Δ𝑋^𝑘_𝑛)^𝑇/(Δ𝑋^𝑘_𝑛)^𝑇Δ𝑔^𝑘_𝑛)+(𝐻^𝑘_𝑛Δ𝑔^𝑘_𝑛(Δ𝑔^𝑘_𝑛)^𝑇𝐻^𝑘_𝑛/(Δ𝑔^𝑘_𝑛)^𝑇𝐻^𝑘_𝑛Δ𝑔^𝑘_𝑛)

   𝑑_𝑛^𝑘+1=−𝐻_𝑛^𝑘+1𝑔_𝑛^𝑘+1,𝑘=𝑘+1

   go to step (3)

(9)  𝐞𝐧𝐝𝐟𝐮𝐧𝐜𝐭𝐢𝐨𝐧