Purpose　-　The　purpose　of　this　paper　is　to　present　a　two-dimensional　(2D)　model　of　the　base　force　element　method　(BFEM)　based　on　the　complementary　energy　principle.　The　study　proposes　a　model　of　the　BFEM　for　arbitrary　mesh　problems.　Design/methodology/approach　-　The　BFEM　uses　the　base　forces　given　by　Gao　(2003)　as　fundamental　variables　to　describe　the　stress　state　of　an　elastic　system.　An　explicit　expression　of　element　compliance　matrix　is　derived　using　the　concept　of　base　forces.　The　detailed　formulations　of　governing　equations　for　the　BFEM　are　given　using　the　Lagrange　multiplier　method.　The　explicit　displacement　expression　of　nodes　is　given.　To　verify　the　2D　model,　a　program　on　the　BFEM　using　MATLAB　language　is　made　and　a　number　of　examples　on　arbitrary　polygonal　meshes　and　aberrant　meshes　are　provided　to　illustrate　the　BFEM.　Findings　-　A　good　agreement　is　obtained　between　the　numerical　and　theoretical　results.　Based　on　the　studies,　it　is　found　that　the　2D　formulation　of　BFEM　with　complementary　energy　principle　provides　reliable　predictions　for　arbitrary　mesh　problems.　Research　limitations/implications　-　Due　to　the　use　of　Lagrange　multiplier　method,　there　are　more　basic　unknowns　in　the　control　equation.　The　proposed　method　will　be　improved　in　the　future.　Practical　implications　-　This　paper　presents　a　new　idea　and　a　new　numerical　method,　and　to　explore　new　ways　to　solve　the　problem　of　arbitrary　meshes.　Originality/value　-　The　paper　presents　a　2D　model　of　the　BFEM　using　the　complementary　energy　principle　for　arbitrary　mesh　problems.