Homogenization　is　an　important　analysis　approach　of　composite　materials　with　structural　hierarchy　and　can　give　the　prediction　of　macroscopic　properties　of　the　composites.　There　are　many　homogenization　theories　and　methods.　The　present　paper　discusses　applications　of　some　homogenization　approaches　including　both　direct　and　mathematical　homogenizations　for　the　analysis　of　anisotropic　composites　with　periodic　microstructures.　The　macroscopic　properties　of　the　composite　are　predicted　by　the　direct　homogenization　and　the　mathematical　homogenisation　method.　The　periodic　boundary　conditions　of　a　representative　volume　element　are　implemented　by　a　transformation　method　of　the　degrees　of　freedom.　The　numerical　results　are　demonstrated　for　two　model　composites.　The　study　shows　that　these　two　homogenization　methods　gave　the　same　results　for　the　macroscopic　elastic　stiffness　of　the　composites　although　they　are　of　different　mathematical　forms　and　different　operation　procedures.