The　solution　accuracy　based　on　finite　element　interpolation　strongly　depends　on　mesh　quality.　Thus,　the　numerical　manifold　method　advocates　that　if　a　finite　element　mesh　is　selected　as　the　mathematical　cover,　the　mesh　should　be　uniform,　where　all　elements　are　congruent　squares　or　isosceles　right　triangles　for　two-dimensional　problems.　However,　the　elements　close　to　the　points　of　stress　concentrations　or　singularities　must　be　smaller　than　those　further　away　from　these　points.　This　leads　to　an　irregular　mesh　in　which　a　few　nodes　of　certain　small　elements　are　hung　on　the　edges　of　adjacent　larger　elements.　This　study　implements　local　refinement　with　arbitrary　k-irregular　meshes　by　selecting　piecewise　polynomials　as　the　local　approximations　over　physical　patches　and　enforcing　appropriate　constraints　on　the　piecewise　polynomials.　All　the　elements　are　ordinary　isoparametric　elements,　and　elements　with　variable　numbers　of　nodes　on　edges　are　not　required.　No　linear　dependency　issue　exists.　Furthermore,　refinement　with　any　number　of　mesh　layers　is　developed　so　that　solution　error　can　be　distributed　as　uniformly　as　possible.　Typical　examples　with　different　singularities　are　designed　to　illustrate　the　effectiveness　of　the　proposed　procedure.