A Novel Mesh Quality Improvement Method for Boundary Elements

Algorithm 2

Line search algorithm.

(1)  function LINE_SEARCH (𝜀₁, 𝜑(𝑓(𝑥), 𝑋⁰_𝑛, ℎ₀,  𝛼))

     ▸  f(𝑥) = objective function

     𝜀₁ = allowable error

     𝜑(f(x), 𝑋⁰_𝑛, ℎ₀, 𝛼) = the function for search region

                 [𝑎₁, 𝑏₁] see below Algorithm 3

(2)  compute 𝜇₁ = 𝑎₁ + 0.382(𝑏₁−𝑎₁) and 𝜈₁ = 𝑎₁ + 0.618(𝑏₁−𝑎₁)

      ▸   𝜇₁,𝜈₁ = initial tentative point

      Set i = 1

      i = iteration number

(3)  if |𝜇_𝑖−𝜈_𝑖| < 𝜀₁ then

(4)   return 𝜆^∗_𝑖=(𝜇_𝑖+𝜈_𝑖)/2 and break

     ▸  𝜆^∗_𝑖 = optimal step size

(5)  else if (f(𝜇_𝑖) < f(𝜈_𝑖)) go to step (7)

(6)          else  𝑓(𝜇_𝑖)≧𝑓(𝜈_𝑖) go to step (8)

(7)  set 𝑎_𝑖+1 = 𝑎_𝑖,  𝑏_𝑖+1 = 𝜈_𝑖,  𝜈_𝑖+1 = 𝜇_𝑖,  f(𝜈_𝑖) = f(𝜇_𝑖)

(8)    compute 𝜇_𝑖+1 = 0.618𝑎_𝑖+1 + 0.382𝑏_𝑖+1 and f(𝜇_𝑖+1)

(9)            i = i+1 go to step (3)