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Crystal-based finite element methods (FEM) are versatile continuum approaches for predicting mechanical properties and deformation-induced crystallographic textures. They can be applied to both, elastic-plastic and elastic problems. The methodology is based on (i) a detailed understanding of the underlying crystal deformation mechanisms and (ii) a number of constitutive material parameters that are often difficult to measure. First principle calculations, that take into account the discrete nature of matter at the atomic scale, are an alternative way to study mechanical properties of single crystals without using empirical parameters. In this study we demonstrate how to combine these two well-established modeling tools, viz., ab initio modeling and crystal mechanical FEM, for an improved approach to design of polycrystalline materials. The combination is based on (i) the determination of basic thermodynamic and elastic parameter trends in metallurgical alloy design using density-functional (DFT) calculations (P. Hohenberg and W. Kohn, Phys. Rev. 136, B864 (1964), W. Kohn and L. J. Sham, Phys. Rev. 140, A1133 (1965) [1, 2], respectively) and (ii) the up-scale transfer of these results into crystal-based finite element simulations which take into account the anisotropic nature of the elastic-plastic deformation of metals. The method is applied to three body-centered cubic (bcc, β) Ti-Nb alloys for bio-medical applications. The study addresses two technological processes, namely, the prediction of texture evolution during cold rolling (elastic-plastic problem) and elastic bending of textured polycrystals (elastic problem).