Recently, mathematics has a wide influence on architectural design and performance. It can effectively help in solving design problems and promoting geometrical shapes efficiency. Consequently, the environmental performance has also widely developed with the involving of advanced mathematical branches and computational software. The main aim of the paper is to study the importance of using mathematics in raising the building's environmental performance. Therefore, the research methodology starts by clarifying the mathematics' branches and their application in architecture, then carries out an analysis for different case studies that used mathematics effectively in developing environmental performance. And lastly, apply one of the mathematical optimization techniques on a virtual prototype to illustrate how mathematics can effectively promote the building's environmental efficiency.

This research focuses on the evolving nature and concepts of digitalcomputational designs and attempts to elucidate the impacts on the architectural design process.It intends to take into account a holistic approach to the integrated design process and some of the key potentialities that emerge from its application in the discipline with the rise of digital design tools in architecture. As a result, the purpose of this research is to examine this new paradigm and provide a conceptual framework based on its models, levels, and systems that aid in the integrated architectural design process.This framework may help architects realize the potential of digital tools by using and thinking of parameterization as a general procedure from the initial stage of form-finding to the final design. Therefore, an awareness of both the potentials and limitations of digital paradigms is imperative in the effective use of these models and in understanding how they will continue to change the architectural profession.It concludes by highlighting the need to sensitize and raise the awareness of professionals and students on the benefits and potentials of the Integrated Design Process enabled by computation.

Existing old buildings were considered as the city's culture reminder. By the time, they could be kept via adaptation for contemporary usages. That adopting considered the context of the environmental, social and economic idea of the prior eras, related to the building's life cycle, that guided by local solutions. This paper started with the adaptation process definition with an evaluation of a number of building's state to bring out the different potential outcomes for discussing the future adaptation possibilities, especially that known as "alterations and extensions". Therefore, by rethinking of the new addition's integration into the heritage buildings will support the heritage value and fit new functions to present innovative design process approaches in the field of heritage preservation's design. By analysing some case studies, the study achieved some criteria for designing the alterations and extensions by making them an effective component in the design of old heritage buildings. And finally applying these principles to demonstrate its efficiency.

Sources of energy are the most important challenges facing the world in this century. The global interest in energy and its relation to the construction sector that consider the most energy consuming sectors, which led to a search for methods of energy conservation. Research assumes a methodology to apply the modern technology in buildings and gaining advantages of developed building materials, outer envelopes, and conserve energy systems to reduce energy waste and increase its efficiency by integrating natural and renewable energy applications. This methodology could be applied to buildings during the design phase to achieve energy rationalization.