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An innovative method is proposed for constructing isoparametric boundary elements to simulate closed surfaces. These elements are named “isoparametric closure elements” and can not only accurately simulate spherical, elliptical, and other closed surface geometries, but also interpolate physical quantities defined over these surfaces. As a result of using the proposed closure elements, each of these surfaces can be discretized into only one element along the circumferential direction. A number of closure elements having 4–26 nodes are investigated to examine the computational error, and three are recommended to be used in the boundary element method (BEM) analysis. These closure elements are applied to BEM analysis of heat conduction and solid mechanics problems. A technique for eliminating singularities involved in boundary integrals over closure elements is also presented. A number of numerical examples will be given to demonstrate the computational accuracy and efficiency of the proposed closure elements.