A Non-linear, Sub-grid Embedded Finite Element Basis for Accurate Monotone Steady Cfd Solutions

A non-linear sub-grid embedded (SGM) nite element basis is derived for generating accurate monotone solutions to a CFD weak statement algorithm. The developed theory connrms that only the second derivative (diiusion) term is appropriate for the SGM construction, which employs element-level static condensation for eeciency and consistency. In comparison to other high resolution methods, advantages of the SGM element formulation include arbitrary (Lagrange) embedding degree, no explicitly added artiicial diiusion term, no ux limiters or switches, improved condition number for the jacobian matrix and excellent algorithm stability. The statically-condensed SGM construction retains linear basis bandwidth, for all problem dimensions, hence exhibits no storage penalty for element or system matrices. Numerical results for 1-D, 2-D and 3-D veriica-tion/benchmark linear and nonlinear convection-diiusion problems in steady state are presented, connrming theoretical predictions for nodally exact and monotone solutions on minimal degree-of-freedom meshes. As a side beneet, commensurate high order accuracy accrues to wall ux prediction on these coarse meshes, via a matrix manipulation on the weak statement algebraic construction. 1 NOMENCLATURE a i expansion coeecient r,r j ,r ij distributed SGM parameter A scalar constant R element right of node j c,c j continuum SGM parameter R statically reduced matrix d dimension of the problem fRg solution residual vector det e transformation matrix determinant < d real d dimensional space D k ] e Lagrange diiusion matrix < + temporal half space D S ] e SGM diiusion matrix Re Reynolds number e nite element s source term f(m) function of m S SGM polynomial degree f j kinematic ux vector S e element matrix assembly operator f v j viscous ux vector t time F, F ij SGM (correlation) function u j ,U velocity vector fFQg Newton residual vector U] e convection matrix g,g i SGM element embedding function jUj absolute value of velocity U fGg nodally distributed SGM vector V e volume (area) of a nite element h,h e ,h ij nite element length measure x j spatial coordinates JAC] jacobian matrix SGM polynomial function of c k Lagrange polynomial degree ,, coeecients in assembly stencil L element left of node j , i physical diiusion coeecient L partial diierential equation operator ij Kronecker delta M] assembled mass (interpolation) matrix t computational time step fN k g Lagrange element basis function of degree k , i local natural coordinate fN S g SGM element basis function of degree S , …