A Novel Mesh Quality Improvement Method for Boundary Elements

Algorithm 3

Interval search algorithm.

(1)  function SEARCH_INTERVAL (f(𝑥), 𝑋⁰_𝑛, ℎ₀, 𝛼)

     ▸ f(x) = objective function

     𝑋⁰_𝑛 = initial coordinate for vertices 𝑋

     ℎ₀ = initial step size

     𝛼 = coefficient larger than 1

(2) compute 𝑓₀ = f(𝑋⁰_𝑛) and Set j = 0

     ▸ j = iteration number

(3)   𝑋_𝑛^𝑗+1 = 𝑋^𝑗_𝑛 + ℎ_𝑗 and compute  f(𝑋_𝑛^𝑗+1)

(4) if  𝑓(𝑋_𝑛^𝑗+1)<𝑓(𝑋^𝑗_𝑛) then go to step (6)

(5) else go to step (7)

(6)   ℎ_𝑗+1 = 𝛼ℎ_𝑗,  𝑋^∗_𝑛 = 𝑋^𝑗_𝑛,  𝑋^𝑗_𝑛 = 𝑋_𝑛^𝑗+1

      f(𝑋^𝑗_𝑛) = f(𝑋_𝑛^𝑗+1),  j = j + 1

(7) if j = 0 then ℎ_𝑗 = −ℎ_𝑗, 𝑋^∗_𝑛 = 𝑋^𝑗_𝑛 go to step (3)

(8) else a = min{𝑋^∗_𝑛, 𝑋_𝑛^𝑗+1},  b = max{𝑋^∗_𝑛, 𝑋_𝑛^𝑗+1}

(9) return 𝑎 and 𝑏